• Syllabus

Topic 7 - Introduction to differential calculus

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Sub sections and their related questions

7.1

• 10M.1.sl.TZ2.15a: State whether f (0) is greater than, less than or equal to f (−2). Give a reason for your answer.
• 10N.2.sl.TZ0.5f: Lines L1 and L2 are parallel, and they are tangents to the graph of f (x) at points A and B...
• 09M.2.sl.TZ1.5g, i: On your graph draw and label the tangent T.
• 11M.2.sl.TZ1.3g: Find the gradient of the tangent to the graph of \(f\) at \(x = 1\).
• 07M.1.sl.TZ0.11c: Draw the tangent to the curved graph for this value of x on the figure, showing clearly the...
• SPM.1.sl.TZ0.5d: where the gradient of the tangent to the curve is positive;
• SPM.1.sl.TZ0.9b: The point \({\text{P}}(3{\text{, }}9)\) lies on the curve \(y = {x^2}\) . Find the gradient of...
• 07N.2.sl.TZ0.5c: Find the coordinates of the point where the tangent to P is parallel to the line L.
• 08N.2.sl.TZ0.5d: Let \({L_1}\) be the tangent to the curve at \(x = 2\). Let \({L_2}\) be a tangent to the curve,...
• 14N.1.sl.TZ0.15b: The gradient of the tangent to the curve is \( - 14\) when \(x = 1\). Find the value of \(a\).
• 15M.2.sl.TZ2.5f: Let \(T\) be the tangent to the graph of \(f\) at \(x =  - 2\). Draw \(T\) on your sketch.
• 16N.2.sl.TZ0.6a: Write down a formula for \(A\), the surface area to be coated.
• 16N.2.sl.TZ0.6b: Express this volume in \({\text{c}}{{\text{m}}^3}\).
• 16N.2.sl.TZ0.6c: Write down, in terms of \(r\) and \(h\), an equation for the volume of this water container.
• 16N.2.sl.TZ0.6e: Find \(\frac{{{\text{d}}A}}{{{\text{d}}r}}\).
• 16N.2.sl.TZ0.6f: Using your answer to part (e), find the value of \(r\) which minimizes \(A\).
• 16N.2.sl.TZ0.6g: Find the value of this minimum area.
• 16N.2.sl.TZ0.6h: Find the least number of cans of water-resistant material that will coat the area in part (g).
• 17M.2.sl.TZ1.6a: Find \(g'(x)\).
• 17M.2.sl.TZ1.6b.i: Show that \(k = 6\).
• 17M.2.sl.TZ1.6b.ii: Find the equation of the tangent to the graph of \(y = g(x)\) at \(x = 2\). Give your answer in...
• 17M.2.sl.TZ1.6c: Use your answer to part (a) and the value of \(k\), to find the \(x\)-coordinates of the...
• 17M.2.sl.TZ1.6d.i: Find \(g’( - 1)\).
• 17M.2.sl.TZ1.6d.ii: Hence justify that \(g\) is decreasing at \(x =  - 1\).
• 17M.2.sl.TZ1.6e: Find the \(y\)-coordinate of the local minimum.
• 17M.2.sl.TZ2.6a: Write down the \(y\)-intercept of the graph.
• 17M.2.sl.TZ2.6b: Find \(f'(x)\).
• 17M.2.sl.TZ2.6c.i: Show that \(a = 8\).
• 17M.2.sl.TZ2.6d.i: Write down the \(x\)-coordinates of these two points;
• 17M.2.sl.TZ2.6e: Write down the range of \(f(x)\).
• 17M.2.sl.TZ2.6f: Write down the number of possible solutions to the equation \(f(x) = 5\).
• 17M.2.sl.TZ2.6g: The equation \(f(x) = m\), where \(m \in \mathbb{R}\), has four solutions. Find the possible...
• 17M.2.sl.TZ2.6c.ii: Find \(f(2)\).
• 17M.2.sl.TZ2.6d.ii: Write down the intervals where the gradient of the graph of \(y = f(x)\) is positive.
• 17N.1.sl.TZ0.14a: Write down the derivative of \(f\).
• 17N.1.sl.TZ0.14b: Find the point on the graph of \(f\) at which the gradient of the tangent is equal to 6.
• 18M.1.sl.TZ1.5a: Write down the coordinates of C, the midpoint of line segment AB.
• 18M.1.sl.TZ1.5b: Find the gradient of the line DC.
• 18M.1.sl.TZ1.5c: Find the equation of the line DC. Write your answer in the form ax + by + d = 0 where a , b and d...
• 18M.1.sl.TZ2.14a: Find f'(x)
• 18M.1.sl.TZ2.14b: Find the gradient of the graph of f at \(x =  - \frac{1}{2}\).
• 18M.1.sl.TZ2.14c: Find the x-coordinate of the point at which the normal to the graph of f has...
• 18M.2.sl.TZ2.6a: Sketch the curve for −1 < x < 3 and −2 < y < 12.
• 18M.2.sl.TZ2.6b: A teacher asks her students to make some observations about the curve. Three students...
• 18M.2.sl.TZ2.6c: Find the value of y when x = 1 .
• 18M.2.sl.TZ2.6d: Find \(\frac{{{\text{dy}}}}{{{\text{dx}}}}\).
• 18M.2.sl.TZ2.6e: Show that the stationary points of the curve are at x = 1 and x = 2.
• 18M.2.sl.TZ2.6f: Given that y = 2x3 − 9x2 + 12x + 2 = k has three solutions, find the possible values of k.
• 16N.2.sl.TZ0.6d: Show that \(A = \pi {r^2} + \frac{{1\,000\,000}}{r}\).

7.2

• 10M.2.sl.TZ1.3b: Write down f ′(x).
• 10N.2.sl.TZ0.5b: Find \(f'(x)\).
• 11N.1.sl.TZ0.14a: Find \(f'(x)\) .
• 12N.1.sl.TZ0.15a: Find f'(x).
• 12N.2.sl.TZ0.5b: Write down g′(x) .
• 10M.2.sl.TZ2.5c: Differentiate A in terms of x.
• 12M.2.sl.TZ1.5d: Find f '(x).
• 12M.1.sl.TZ2.13a: Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).
• 12M.2.sl.TZ2.5c: Find \( \frac{{\text{d}V}}{{\text{d}x}}\).
• 09N.1.sl.TZ0.6a: Find \(f'(x)\).
• 11N.2.sl.TZ0.4b: Find  \(f'(x)\) .
• 09N.2.sl.TZ0.5B, b, i: The gradient of the curve \(y = p{x^2} + qx - 4\) at the point (2, –10) is 1. Find...
• 09M.1.sl.TZ1.15b: Sarah wishes to draw the tangent to \(f (x) = x^4\) parallel to L. Write down \(f ′(x)\).
• 09M.2.sl.TZ1.5a: Differentiate \(f (x)\) with respect to \(x\).
• 11M.2.sl.TZ1.3c: Find \(f'(x)\) .
• 09M.2.sl.TZ2.5a: Write down an expression for \(f ′(x)\).
• 09M.1.sl.TZ2.11a: Find \(f ′(x)\).
• 11M.1.sl.TZ2.11a: Differentiate \(f(x)\) with respect to \(x\) .
• 11M.1.sl.TZ2.11b: Differentiate \(g(x)\) with respect to \(x\) .
• 13M.1.sl.TZ1.15a: Find \(f ' (x) \).
• 13M.2.sl.TZ1.4b: Find \(f ′(x)\).
• 11M.2.sl.TZ2.5b: Find \(f'(x)\) .
• 13M.1.sl.TZ2.11a: Find \(f ' (x) \).
• 13M.2.sl.TZ2.5b: Find the derivative of \(y = \frac{{ - {x^2}}}{{10}} + \frac{{27}}{2}x\).
• 07M.1.sl.TZ0.11a: (i) Differentiate \(f_1 (x) \) with respect to x. (ii) Differentiate \(f_2 (x) \) with respect...
• SPM.1.sl.TZ0.9a: Write down \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).
• SPM.1.sl.TZ0.14a: Write down \(f'(x)\) .
• SPM.1.sl.TZ0.15b: Find the number of machines that should be made and sold each month to maximize \(P(x)\) .
• 07M.2.sl.TZ0.3i.b: Find \(f ′(x)\).
• 07M.2.sl.TZ0.3ii.b: Find \(\frac{{dy}}{{dx}}\).
• SPM.2.sl.TZ0.6e: Write down \(\frac{{{\text{d}}A}}{{{\text{d}}r}}\).
• 07N.1.sl.TZ0.15a: Find \(f ′(x)\) .
• 07N.2.sl.TZ0.5b: Differentiate \(f (x)\) .
• 08N.2.sl.TZ0.5c: Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) .
• 08M.1.sl.TZ1.3a: Find \(f'(x)\).
• 08M.2.sl.TZ1.5ii.d: Find \(\frac{{{\text{d}}V}}{{{\text{d}}x}}\).
• 08M.1.sl.TZ2.12a: Find \(f'(x)\).
• 08M.1.sl.TZ2.12b: Find \(f''(x)\).
• 08M.2.sl.TZ2.4ii.a: Find \(C'(x)\).
• 14M.1.sl.TZ2.15a: Write down the derivative of the function.
• 14M.2.sl.TZ2.5e: The parcel is tied up using a length of string that fits exactly around the parcel, as shown in...
• 13N.1.sl.TZ0.9b: Differentiate \(f(x) = x(2{x^3} - 1)\).
• 13N.2.sl.TZ0.4b: Find \(f'(x)\).
• 14M.1.sl.TZ1.10a: Write down \(f'(x)\).
• 14M.1.sl.TZ1.15a: Write down \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).
• 14M.2.sl.TZ1.6d: The volume of the lobster trap is \(0.75{\text{ }}{{\text{m}}^{\text{3}}}\). Find...
• 14N.1.sl.TZ0.15a: Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).
• 14N.2.sl.TZ0.3e: A company designs cone-shaped tents to resemble the traditional tepees. These cone-shaped tents...
• 15M.1.sl.TZ1.15b: Find the value of \(x\) that makes the volume a maximum.
• 15M.2.sl.TZ1.5a: Write down \(f'(x)\).
• 15M.2.sl.TZ2.5b: Find \(f'(x)\).
• 16M.1.sl.TZ1.11a: Consider the function \(f(x) = a{x^2} + c\). Find \(f'(x)\)  
• 16M.1.sl.TZ1.11b: Point \({\text{A}}( - 2,\,5)\)  lies on the graph of \(y = f(x)\) . The gradient of the tangent...
• 16M.1.sl.TZ1.11c: Find the value of \(c\) .
• 16M.1.sl.TZ1.15a: A company sells fruit juices in cylindrical cans, each of which has a volume of...
• 16M.1.sl.TZ1.15b: Calculate the value of \(r\) that minimizes the surface area of a can.
• 16M.2.sl.TZ1.3a: A distress flare is fired into the air from a ship at sea. The height, \(h\) , in metres, of the...
• 16M.2.sl.TZ1.3b: Find the height of the flare \(15\) seconds after it was fired.
• 16M.2.sl.TZ1.3c: The flare fell into the sea \(k\) seconds after it was fired. Find the value of \(k\) .
• 16M.2.sl.TZ1.3d: Find \(h'\,(t)\,.\)
• 16M.2.sl.TZ1.3e: i)     Show that the flare reached its maximum height \(40\) seconds after being fired. ii)  ...
• 16M.2.sl.TZ1.3f: The nearest coastguard can see the flare when its height is more than \(40\) metres above sea...
• 16N.1.sl.TZ0.14a: Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).
• 16N.1.sl.TZ0.14b: Find the coordinates of P.
• 16M.1.sl.TZ2.15a: Consider the function \(f(x) = {x^3} - 3{x^2} + 2x + 2\) . Part of the graph of \(f\) is shown...
• 16M.1.sl.TZ2.15b: There are two points at which the gradient of the graph of \(f\) is \(11\). Find...
• 16N.2.sl.TZ0.6a: Write down a formula for \(A\), the surface area to be coated.
• 16N.2.sl.TZ0.6b: Express this volume in \({\text{c}}{{\text{m}}^3}\).
• 16N.2.sl.TZ0.6c: Write down, in terms of \(r\) and \(h\), an equation for the volume of this water container.
• 16N.2.sl.TZ0.6e: Find \(\frac{{{\text{d}}A}}{{{\text{d}}r}}\).
• 16N.2.sl.TZ0.6f: Using your answer to part (e), find the value of \(r\) which minimizes \(A\).
• 16N.2.sl.TZ0.6g: Find the value of this minimum area.
• 16N.2.sl.TZ0.6h: Find the least number of cans of water-resistant material that will coat the area in part (g).
• 16M.2.sl.TZ2.5a: Hugo is given a rectangular piece of thin cardboard, \(16\,{\text{cm}}\) by \(10\,{\text{cm}}\)....
• 16M.2.sl.TZ2.5b: (i)     State whether \(x\) can have a value of \(5\). Give a reason for your answer. (ii)  ...
• 16M.2.sl.TZ2.5c: Show that the volume, \(V\,{\text{c}}{{\text{m}}^3}\), of this tray is given...
• 16M.2.sl.TZ2.5d: Find \(\frac{{dV}}{{dx}}.\)
• 16M.2.sl.TZ2.5e: Using your answer from part (d), find the value of \(x\) that maximizes the volume of the tray.
• 16M.2.sl.TZ2.5f: Calculate the maximum volume of the tray.
• 16M.2.sl.TZ2.5g: Sketch the graph of \(V = 4{x^3} - 51{x^2} + 160x\) , for the possible values of \(x\) found...
• 17M.2.sl.TZ1.6a: Find \(g'(x)\).
• 17M.2.sl.TZ1.6b.i: Show that \(k = 6\).
• 17M.2.sl.TZ1.6b.ii: Find the equation of the tangent to the graph of \(y = g(x)\) at \(x = 2\). Give your answer in...
• 17M.2.sl.TZ1.6d.i: Find \(g’( - 1)\).
• 17M.2.sl.TZ1.6d.ii: Hence justify that \(g\) is decreasing at \(x =  - 1\).
• 17M.2.sl.TZ1.6e: Find the \(y\)-coordinate of the local minimum.
• 17M.2.sl.TZ2.6a: Write down the \(y\)-intercept of the graph.
• 17M.2.sl.TZ2.6b: Find \(f'(x)\).
• 17M.2.sl.TZ2.6c.i: Show that \(a = 8\).
• 17M.2.sl.TZ2.6d.i: Write down the \(x\)-coordinates of these two points;
• 17M.2.sl.TZ2.6e: Write down the range of \(f(x)\).
• 17M.2.sl.TZ2.6f: Write down the number of possible solutions to the equation \(f(x) = 5\).
• 17M.2.sl.TZ2.6g: The equation \(f(x) = m\), where \(m \in \mathbb{R}\), has four solutions. Find the possible...
• 17M.2.sl.TZ2.6c.ii: Find \(f(2)\).
• 17M.2.sl.TZ2.6d.ii: Write down the intervals where the gradient of the graph of \(y = f(x)\) is positive.
• 17N.1.sl.TZ0.14a: Write down the derivative of \(f\).
• 17N.1.sl.TZ0.14b: Find the point on the graph of \(f\) at which the gradient of the tangent is equal to 6.
• 17N.2.sl.TZ0.5a: Find the exact value of each of the zeros of \(f\).
• 17N.2.sl.TZ0.5b.i: Expand the expression for \(f(x)\).
• 17N.2.sl.TZ0.5b.ii: Find \(f’(x)\).
• 17N.2.sl.TZ0.5c: Use your answer to part (b)(ii) to find the values of \(x\) for which \(f\) is increasing.
• 17N.2.sl.TZ0.5d: Draw the graph of \(f\) for \( - 3 \leqslant x \leqslant 3\) and...
• 17N.2.sl.TZ0.5e: Write down the coordinates of the point of intersection.
• 18M.2.sl.TZ1.6a: Write down the height of the cylinder.
• 18M.2.sl.TZ1.6b: Find the total volume of the trash can.
• 18M.2.sl.TZ1.6c: Find the height of the cylinder, h , of the new trash can, in terms of r.
• 18M.2.sl.TZ1.6d: Show that the volume, V cm3 , of the new trash can is given by \(V = 110\pi {r^3}\).
• 18M.2.sl.TZ1.6e: Using your graphic display calculator, find the value of r which maximizes the value of V.
• 18M.2.sl.TZ1.6f: The designer claims that the new trash can has a capacity that is at least 40% greater than the...
• 18M.1.sl.TZ2.14a: Find f'(x)
• 18M.1.sl.TZ2.14b: Find the gradient of the graph of f at \(x =  - \frac{1}{2}\).
• 18M.1.sl.TZ2.14c: Find the x-coordinate of the point at which the normal to the graph of f has...
• 18M.2.sl.TZ2.6a: Sketch the curve for −1 < x < 3 and −2 < y < 12.
• 18M.2.sl.TZ2.6b: A teacher asks her students to make some observations about the curve. Three students...
• 18M.2.sl.TZ2.6c: Find the value of y when x = 1 .
• 18M.2.sl.TZ2.6d: Find \(\frac{{{\text{dy}}}}{{{\text{dx}}}}\).
• 18M.2.sl.TZ2.6e: Show that the stationary points of the curve are at x = 1 and x = 2.
• 18M.2.sl.TZ2.6f: Given that y = 2x3 − 9x2 + 12x + 2 = k has three solutions, find the possible values of k.
• 16N.2.sl.TZ0.6d: Show that \(A = \pi {r^2} + \frac{{1\,000\,000}}{r}\).

7.3

• 10M.1.sl.TZ2.15b: The point P(−2, 3) lies on the graph of f (x). Write down the equation of the tangent to the...
• 10M.2.sl.TZ1.3e: Let P be the point where the graph of f (x) intersects the y axis. Find the gradient of the...
• 10M.2.sl.TZ1.3f: The line, L, is the tangent to the graph of f (x) at P. Find the equation of L in the form y =...
• 10M.2.sl.TZ1.3h: There is a second point, Q, on the curve at which the tangent to f (x) is parallel to...
• 10N.2.sl.TZ0.5c: Find the gradient of the graph of f (x) at the point where x = 1.
• 10N.2.sl.TZ0.5f: Lines L1 and L2 are parallel, and they are tangents to the graph of f (x) at points A and B...
• 12N.2.sl.TZ0.5c: The line T is the tangent to the graph of y = g(x) at the point where x = 1. The gradient of T is...
• 12N.2.sl.TZ0.5d: The line T is the tangent to the graph of y = g(x) at the point where x = 1. The gradient of T is...
• 12M.2.sl.TZ1.5f: Explain what f '(−1) represents.
• 12M.2.sl.TZ1.5g: Find the equation of the tangent to the graph of f (x) at the point where x is –1.
• 12M.1.sl.TZ2.13b: The equation of the line L is \(6x + 2y = -1\). Find the x-coordinate of the point on the curve...
• 09N.1.sl.TZ0.6b: Find the value of \(f'( - 3)\).
• 11N.2.sl.TZ0.4d: Find the gradient of the tangent to \(y = f (x)\) at the point \({\text{A}}(1{\text{, }}8)\) .
• 11N.2.sl.TZ0.4e: The line, \(L\), passes through the point A and is perpendicular to the tangent at A. Write...
• 11N.2.sl.TZ0.4f: The line, \(L\) , passes through the point A and is perpendicular to the tangent at A. Find the...
• 09M.1.sl.TZ1.15c, i: Find the x coordinate of the point at which the tangent must be drawn.
• 09M.2.sl.TZ1.5b: Calculate \(f ′(x)\) when \(x = 1\).
• 11M.2.sl.TZ1.3g: Find the gradient of the tangent to the graph of \(f\) at \(x = 1\).
• 11M.2.sl.TZ1.3h: There is a second point on the graph of \(f\) at which the tangent is parallel to the tangent at...
• 09M.2.sl.TZ2.5d, i: Let T be the tangent to the graph of f at P. Show that the gradient of T is –7.
• 09M.1.sl.TZ2.11c: Let L be the line with equation y = 3x + 2. Let P be a point on the curve of f. At P, the...
• 11M.1.sl.TZ2.11c: Calculate the value of \(x\) for which the gradients of the two graphs are the same.
• 11M.1.sl.TZ2.11d: Draw the tangent to the parabola at the point with the value of \(x\) found in part (c).
• 13M.2.sl.TZ1.4h: L is the tangent to the graph of the function \(y = f (x)\), at the point on the graph with the...
• 13M.2.sl.TZ1.4i: Find the equation of L. Give your answer in the form \(y = mx + c\).
• 11M.2.sl.TZ2.5c: Find the gradient of the graph of the function at \(x = - 1\).
• 13M.1.sl.TZ2.11b: The gradient of the curve at point A is 35. Find the x-coordinate of point A.
• 07M.1.sl.TZ0.11b: Calculate the value of x for which the gradient of the two graphs is the same.
• SPM.1.sl.TZ0.5d: where the gradient of the tangent to the curve is positive;
• SPM.1.sl.TZ0.9b: The point \({\text{P}}(3{\text{, }}9)\) lies on the curve \(y = {x^2}\) . Find the gradient of...
• SPM.1.sl.TZ0.9c: The point \({\text{P}}(3{\text{, }}9)\) lies on the curve \(y = {x^2}\) . Find the equation of...
• SPM.1.sl.TZ0.14b: Find the equation of the tangent to the graph of \(y = f(x)\) at \((1{\text{, }}3)\) .
• 07N.2.sl.TZ0.1ii.c: Find the value of the gradient of the curve where \(x = 1.7\) .
• 07N.2.sl.TZ0.5d: Find the coordinates of the point where the tangent to P is perpendicular to the line L.
• 07N.2.sl.TZ0.5e: Find (i) the gradient of the tangent to P at the point with coordinates (2, − 6). (ii) the...
• 07N.2.sl.TZ0.5g: Find the coordinates of the vertex of P and state the gradient of the curve at this point.
• 08N.2.sl.TZ0.5d: Let \({L_1}\) be the tangent to the curve at \(x = 2\). Let \({L_2}\) be a tangent to the curve,...
• 08N.2.sl.TZ0.5e: It is known that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} > 0\) for \(x < - 2\) and...
• 08M.1.sl.TZ1.3b: Write down the value of \(f'(2)\).
• 08M.1.sl.TZ1.3c: Find the equation of the tangent to the curve of \(y = f(x)\) at the point \((2{\text{, }}3)\).
• 08M.1.sl.TZ2.12c: Find the equation of the tangent to the curve of \(f\) at the point \((1{\text{, }}1.5)\).
• 08M.1.sl.TZ2.15a: Write down the equation of the tangent to the graph of \(f(x)\) at \({\text{P}}\).
• 09M.2.sl.TZ1.5e, iii: The graph of f has a local minimum at point P. Let T be the tangent to the graph of f at...
• 09M.2.sl.TZ2.5d, ii: Let T be the tangent to the graph of f at P. Find the equation of T.
• 09N.2.sl.TZ0.5B, b, ii: The gradient of the curve \(y = p{x^2} + qx - 4\) at the point (2, –10) is 1. Hence, find a...
• 14M.1.sl.TZ2.13d: Draw the tangent to the curve at \(x = 1\) on the graph.
• 14M.1.sl.TZ2.13e: Write down the equation of the tangent at \(x = 1\).
• 13N.2.sl.TZ0.4f: Let \(T\) be the tangent to the graph of the function \(f(x)\) at the point \((2, –12)\). Find...
• 14M.1.sl.TZ1.10b: Point \({\text{P}}(2,6)\) lies on the graph of \(f\). Find the gradient of the tangent to the...
• 14M.1.sl.TZ1.10c: Point \({\text{P}}(2,16)\) lies on the graph of \(f\). Find the equation of the normal to the...
• 14N.1.sl.TZ0.15b: The gradient of the tangent to the curve is \( - 14\) when \(x = 1\). Find the value of \(a\).
• 15M.2.sl.TZ1.5e: The graph of \(y = f(x)\) has a local minimum point at \(x = 4\). Find the equation of the...
• 15M.2.sl.TZ2.5c: Find the gradient of the graph of \(f\) at \(x =  - 2\).
• 15M.2.sl.TZ2.5d: Let \(T\) be the tangent to the graph of \(f\) at \(x =  - 2\). Write down the equation of \(T\).
• 15M.2.sl.TZ2.5f: Let \(T\) be the tangent to the graph of \(f\) at \(x =  - 2\). Draw \(T\) on your sketch.
• 16M.1.sl.TZ1.7a: The equation of line \({L_1}\) is \(y = 2.5x + k\). Point \({\text{A}}\) \(\,(3,\, - 2)\) lies on...
• 16M.1.sl.TZ1.7b: The line \({L_2}\) is perpendicular to \({L_1}\) and intersects \({L_1}\) at point...
• 16M.1.sl.TZ1.7c: Find the equation of \({L_2}\). Give your answer in the form \(y = mx + c\) .
• 16M.1.sl.TZ1.7d: Write your answer to part (c) in the form \(ax + by + d = 0\)  where \(a\), \(b\) and...
• 16M.1.sl.TZ1.11a: Consider the function \(f(x) = a{x^2} + c\). Find \(f'(x)\)  
• 16M.1.sl.TZ1.11b: Point \({\text{A}}( - 2,\,5)\)  lies on the graph of \(y = f(x)\) . The gradient of the tangent...
• 16M.1.sl.TZ1.11c: Find the value of \(c\) .
• 16M.2.sl.TZ1.6a: A function, \(f\) , is given by \[f(x) = 4 \times {2^{ - x}} + 1.5x - 5.\] Calculate \(f(0)\)  
• 16M.2.sl.TZ1.6b: Use your graphic display calculator to solve \(f(x) = 0.\)
• 16M.2.sl.TZ1.6c: Sketch the graph of \(y = f(x)\) for \( - 2 \leqslant x \leqslant 6\) and...
• 16M.2.sl.TZ1.6d: The function \(f\) is the derivative of a function \(g\) . It is known that \(g(1) = 3.\) i)    ...
• 16N.1.sl.TZ0.14a: Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).
• 16N.1.sl.TZ0.14b: Find the coordinates of P.
• 16M.1.sl.TZ2.12a: The equation of the straight line \({L_1}\) is \(y = 2x - 3.\) Write down the \(y\)-intercept of...
• 16M.1.sl.TZ2.12b: Write down the gradient of \({L_1}\) .
• 16M.1.sl.TZ2.12c: The line \({L_2}\) is parallel to \({L_1}\) and passes through the point \((0,\,\,3)\) . Write...
• 16M.1.sl.TZ2.12d: The line \({L_3}\) is perpendicular to \({L_1}\) and passes through the point \(( -...
• 16M.1.sl.TZ2.12e: Find the equation of \({L_3}\) . Give your answer in the form \(ax + by + d = 0\) , where \(a\) ,...
• 16M.1.sl.TZ2.15a: Consider the function \(f(x) = {x^3} - 3{x^2} + 2x + 2\) . Part of the graph of \(f\) is shown...
• 16M.1.sl.TZ2.15b: There are two points at which the gradient of the graph of \(f\) is \(11\). Find...
• 17M.1.sl.TZ1.11a: Write down the gradient of \({L_1}\).
• 17M.1.sl.TZ1.11b: Find the \(y\)-coordinate of P.
• 17M.1.sl.TZ1.11c: Determine the equation of \({L_2}\). Give your answer in the form \(ax + by + d = 0\), where...
• 17M.1.sl.TZ2.13a: Write down the value of \(f(1)\).
• 17M.1.sl.TZ2.13b: Find the equation of \(N\). Give your answer in the form \(ax + by + d = 0\) where \(a\), \(b\),...
• 17M.1.sl.TZ2.13c: Draw the line \(N\) on the diagram above.
• 17M.2.sl.TZ1.6a: Find \(g'(x)\).
• 17M.2.sl.TZ1.6b.i: Show that \(k = 6\).
• 17M.2.sl.TZ1.6b.ii: Find the equation of the tangent to the graph of \(y = g(x)\) at \(x = 2\). Give your answer in...
• 17M.2.sl.TZ1.6c: Use your answer to part (a) and the value of \(k\), to find the \(x\)-coordinates of the...
• 17M.2.sl.TZ1.6d.i: Find \(g’( - 1)\).
• 17M.2.sl.TZ1.6d.ii: Hence justify that \(g\) is decreasing at \(x =  - 1\).
• 17M.2.sl.TZ1.6e: Find the \(y\)-coordinate of the local minimum.
• 17M.2.sl.TZ2.6a: Write down the \(y\)-intercept of the graph.
• 17M.2.sl.TZ2.6b: Find \(f'(x)\).
• 17M.2.sl.TZ2.6c.i: Show that \(a = 8\).
• 17M.2.sl.TZ2.6d.i: Write down the \(x\)-coordinates of these two points;
• 17M.2.sl.TZ2.6e: Write down the range of \(f(x)\).
• 17M.2.sl.TZ2.6f: Write down the number of possible solutions to the equation \(f(x) = 5\).
• 17M.2.sl.TZ2.6g: The equation \(f(x) = m\), where \(m \in \mathbb{R}\), has four solutions. Find the possible...
• 17M.2.sl.TZ2.6c.ii: Find \(f(2)\).
• 17M.2.sl.TZ2.6d.ii: Write down the intervals where the gradient of the graph of \(y = f(x)\) is positive.
• 17N.1.sl.TZ0.2a: Find the coordinates of M.
• 17N.1.sl.TZ0.2b: Find the gradient of \({L_1}\).
• 17N.1.sl.TZ0.2c.i: Write down the gradient of \({L_2}\).
• 17N.1.sl.TZ0.2c.ii: Write down, in the form \(y = mx + c\), the equation of \({L_2}\).
• 17N.1.sl.TZ0.14a: Write down the derivative of \(f\).
• 17N.1.sl.TZ0.14b: Find the point on the graph of \(f\) at which the gradient of the tangent is equal to 6.
• 18M.1.sl.TZ1.5a: Write down the coordinates of C, the midpoint of line segment AB.
• 18M.1.sl.TZ1.5b: Find the gradient of the line DC.
• 18M.1.sl.TZ1.5c: Find the equation of the line DC. Write your answer in the form ax + by + d = 0 where a , b and d...
• 18M.2.sl.TZ1.4a: Find the value of k.
• 18M.2.sl.TZ1.4b: Using your value of k , find f ′(x).
• 18M.2.sl.TZ1.4c: Use your answer to part (b) to show that the minimum value of f(x) is −22 .
• 18M.2.sl.TZ1.4d: Write down the two values of x which satisfy f (x) = 0.
• 18M.2.sl.TZ1.4e: Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.Clearly indicate the minimum...
• 18M.1.sl.TZ2.14a: Find f'(x)
• 18M.1.sl.TZ2.14b: Find the gradient of the graph of f at \(x =  - \frac{1}{2}\).
• 18M.1.sl.TZ2.14c: Find the x-coordinate of the point at which the normal to the graph of f has...
• 18M.2.sl.TZ2.6a: Sketch the curve for −1 < x < 3 and −2 < y < 12.
• 18M.2.sl.TZ2.6b: A teacher asks her students to make some observations about the curve. Three students...
• 18M.2.sl.TZ2.6c: Find the value of y when x = 1 .
• 18M.2.sl.TZ2.6d: Find \(\frac{{{\text{dy}}}}{{{\text{dx}}}}\).
• 18M.2.sl.TZ2.6e: Show that the stationary points of the curve are at x = 1 and x = 2.
• 18M.2.sl.TZ2.6f: Given that y = 2x3 − 9x2 + 12x + 2 = k has three solutions, find the possible values of k.

7.4

• 10M.1.sl.TZ2.15a: State whether f (0) is greater than, less than or equal to f (−2). Give a reason for your answer.
• 11N.1.sl.TZ0.14b: Find the interval of \(x\) for which \(f(x)\) is decreasing.
• 12N.2.sl.TZ0.5h: Write down the interval for which g(x) is increasing in the domain 0 < x < 5 .
• 12M.2.sl.TZ1.5j: P and Q are points on the curve such that the tangents to the curve at these points are...
• 09M.2.sl.TZ1.5c: Use your answer to part (b) to decide whether the function, \(f\) , is increasing or decreasing...
• 11M.2.sl.TZ2.5d: Using your answer to part (c), decide whether the function \(f(x)\) is increasing or decreasing...
• SPM.1.sl.TZ0.5e: where \(f(x) > 0\) and \(f'(x) < 0\) .
• 07M.2.sl.TZ0.3i.d: Write down all intervals in the given domain for which \(f (x)\) is increasing.
• 08N.2.sl.TZ0.5e: It is known that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} > 0\) for \(x < - 2\) and...
• 08M.1.sl.TZ2.15b: State whether \(f(4)\) is greater than, equal to or less than \(f(2)\).
• 08M.1.sl.TZ2.15c: Given that \(f(x)\) is increasing for \(4 \leqslant x < 7\), what can you say about the point...
• 09M.2.sl.TZ2.5c, ii: There is a local minimum at the point Q. Find the set of values of x for which the function is...
• 14M.1.sl.TZ2.13c: Write down the interval where \(f'(x) < 0\).
• 15M.2.sl.TZ1.5h: The graph of \(y = f(x)\) has a local minimum point at \(x = 4\). State the values of \(x\) for...
• 15M.2.sl.TZ2.5e: Let \(T\) be the tangent to the graph of \(f\) at \(x =  - 2\). Sketch the graph of \(f\) for...
• 17M.2.sl.TZ1.6a: Find \(g'(x)\).
• 17M.2.sl.TZ1.6b.i: Show that \(k = 6\).
• 17M.2.sl.TZ1.6b.ii: Find the equation of the tangent to the graph of \(y = g(x)\) at \(x = 2\). Give your answer in...
• 17M.2.sl.TZ1.6c: Use your answer to part (a) and the value of \(k\), to find the \(x\)-coordinates of the...
• 17M.2.sl.TZ1.6d.i: Find \(g’( - 1)\).
• 17M.2.sl.TZ1.6d.ii: Hence justify that \(g\) is decreasing at \(x =  - 1\).
• 17M.2.sl.TZ1.6e: Find the \(y\)-coordinate of the local minimum.
• 17M.2.sl.TZ2.6a: Write down the \(y\)-intercept of the graph.
• 17M.2.sl.TZ2.6b: Find \(f'(x)\).
• 17M.2.sl.TZ2.6c.i: Show that \(a = 8\).
• 17M.2.sl.TZ2.6d.i: Write down the \(x\)-coordinates of these two points;
• 17M.2.sl.TZ2.6e: Write down the range of \(f(x)\).
• 17M.2.sl.TZ2.6f: Write down the number of possible solutions to the equation \(f(x) = 5\).
• 17M.2.sl.TZ2.6g: The equation \(f(x) = m\), where \(m \in \mathbb{R}\), has four solutions. Find the possible...
• 17M.2.sl.TZ2.6c.ii: Find \(f(2)\).
• 17M.2.sl.TZ2.6d.ii: Write down the intervals where the gradient of the graph of \(y = f(x)\) is positive.
• 17N.2.sl.TZ0.5a: Find the exact value of each of the zeros of \(f\).
• 17N.2.sl.TZ0.5b.i: Expand the expression for \(f(x)\).
• 17N.2.sl.TZ0.5b.ii: Find \(f’(x)\).
• 17N.2.sl.TZ0.5c: Use your answer to part (b)(ii) to find the values of \(x\) for which \(f\) is increasing.
• 17N.2.sl.TZ0.5d: Draw the graph of \(f\) for \( - 3 \leqslant x \leqslant 3\) and...
• 17N.2.sl.TZ0.5e: Write down the coordinates of the point of intersection.
• 18M.2.sl.TZ2.6a: Sketch the curve for −1 < x < 3 and −2 < y < 12.
• 18M.2.sl.TZ2.6b: A teacher asks her students to make some observations about the curve. Three students...
• 18M.2.sl.TZ2.6c: Find the value of y when x = 1 .
• 18M.2.sl.TZ2.6d: Find \(\frac{{{\text{dy}}}}{{{\text{dx}}}}\).
• 18M.2.sl.TZ2.6e: Show that the stationary points of the curve are at x = 1 and x = 2.
• 18M.2.sl.TZ2.6f: Given that y = 2x3 − 9x2 + 12x + 2 = k has three solutions, find the possible values of k.

7.5

• 10M.1.sl.TZ2.15c: The point P(−2, 3) lies on the graph of f (x). From the information given about f ′(x), state...
• 10M.2.sl.TZ1.3c: Find the value of the local maximum of y = f (x).
• 10N.2.sl.TZ0.5d: (i) Use f '(x) to find the x-coordinate of M and of N. (ii) Hence or otherwise write down the...
• 12N.1.sl.TZ0.15b: Find using your answer to part (a) the x-coordinate of (i) the local maximum point; (ii) the...
• 12N.2.sl.TZ0.5g: Using your graphic display calculator find the coordinates of the local minimum point of y = g(x) .
• 10M.2.sl.TZ2.5d: Find the value of x that makes A a minimum.
• 12M.2.sl.TZ2.5d: Find the value of x for which V is a maximum.
• 12M.2.sl.TZ2.5e: Find the maximum volume of the container.
• 12M.2.sl.TZ2.5f: Find the length and height of the container for which the volume is a maximum.
• 09N.1.sl.TZ0.6c: Find the value of \(x\) for which \(f'(x) = 0\).
• 09N.2.sl.TZ0.5A, f: Use Differential Calculus to verify that your answer to (e) is correct.
• 09M.2.sl.TZ1.5d: Solve the equation \(f ′(x) = 0\).
• 09M.2.sl.TZ1.5e, i: The graph of f has a local minimum at point P. Let T be the tangent to the graph of f at...
• 11M.2.sl.TZ1.3e: Write down the coordinates of the local maximum point on the graph of \(f\) .
• 13M.1.sl.TZ1.15c: The function has a local maximum at x = −2. Calculate the value of a.
• 13M.2.sl.TZ1.4c: Use your answer to part (b) to show that the x-coordinate of the local minimum point of the graph...
• 11M.2.sl.TZ2.5f: \({{\text{P}}_1}\) is the local maximum point and \({{\text{P}}_2}\) is the local minimum point...
• 13M.2.sl.TZ2.5c: Use the answer in part (b) to determine if A (75, 450) is the point furthest north on the track...
• SPM.1.sl.TZ0.5a: that are local maximum points;
• SPM.1.sl.TZ0.15b: Find the number of machines that should be made and sold each month to maximize \(P(x)\) .
• 07M.2.sl.TZ0.3i.c: Using your graphic display calculator or otherwise, write down the coordinates of any point where...
• 07M.2.sl.TZ0.3ii.c: (i) Use your answer to part (b) to calculate the horizontal distance the ball has travelled from...
• SPM.2.sl.TZ0.6f: (i)     Find the value of \(r\) that minimizes the total external surface area of the wastepaper...
• 07N.1.sl.TZ0.15b: The function \(f (x)\) has a local maximum at the point where \(x = −1\). Find the value of a.
• 07N.2.sl.TZ0.5g: Find the coordinates of the vertex of P and state the gradient of the curve at this point.
• 08M.2.sl.TZ1.5ii.e: (i)     Hence find the value of \(x\) and of \(y\) required to make the volume of the box a...
• 08M.2.sl.TZ2.4ii.b: Show that the cost per person is a minimum when \(10\) people are invited to the party.
• 09M.2.sl.TZ1.5e, ii: The graph of f has a local minimum at point P. Let T be the tangent to the graph of f at...
• 14M.1.sl.TZ2.13a: Label the local maximum as \({\text{A}}\) on the graph.
• 14M.1.sl.TZ2.13b: Label the local minimum as B on the graph.
• 14M.1.sl.TZ2.15b: Use your graphic display calculator to find the coordinates of the local minimum point of...
• 13N.1.sl.TZ0.9c: Find the \(x\)-coordinate of the local minimum of the curve \(y = f(x)\).
• 13N.2.sl.TZ0.4c: The graph of the function \(f(x)\) has a local minimum at the point where \(x =  - 2\). Using...
• 13N.2.sl.TZ0.4e: The graph of the function \(f(x)\) has a local minimum at the point where \(x =  - 2\). Write...
• 14M.1.sl.TZ1.15b: The curve has a local minimum at the point where \(x = 2\). Find the value of \(k\).
• 14M.1.sl.TZ1.15c: The curve has a local minimum at the point where \(x = 2\). Find the value of \(y\) at this...
• 15M.1.sl.TZ1.15b: Find the value of \(x\) that makes the volume a maximum.
• 15M.2.sl.TZ1.5c: The graph of \(y = f(x)\) has a local minimum point at \(x = 4\). Find \(f(2)\).
• 15M.2.sl.TZ1.5d: The graph of \(y = f(x)\) has a local minimum point at \(x = 4\). Find \(f'(2)\)
• 15M.2.sl.TZ1.5e: The graph of \(y = f(x)\) has a local minimum point at \(x = 4\). Find the equation of the...
• 15M.2.sl.TZ1.5f: The graph of \(y = f(x)\) has a local minimum point at \(x = 4\). Sketch the graph of...
• 15M.2.sl.TZ1.5b: The graph of \(y = f(x)\) has a local minimum point at \(x = 4\). Show that \(k = 3\).
• 16M.2.sl.TZ1.3a: A distress flare is fired into the air from a ship at sea. The height, \(h\) , in metres, of the...
• 16M.2.sl.TZ1.3b: Find the height of the flare \(15\) seconds after it was fired.
• 16M.2.sl.TZ1.3c: The flare fell into the sea \(k\) seconds after it was fired. Find the value of \(k\) .
• 16M.2.sl.TZ1.3d: Find \(h'\,(t)\,.\)
• 16M.2.sl.TZ1.3e: i)     Show that the flare reached its maximum height \(40\) seconds after being fired. ii)  ...
• 16M.2.sl.TZ1.3f: The nearest coastguard can see the flare when its height is more than \(40\) metres above sea...
• 16N.2.sl.TZ0.6a: Write down a formula for \(A\), the surface area to be coated.
• 16N.2.sl.TZ0.6b: Express this volume in \({\text{c}}{{\text{m}}^3}\).
• 16N.2.sl.TZ0.6c: Write down, in terms of \(r\) and \(h\), an equation for the volume of this water container.
• 16N.2.sl.TZ0.6e: Find \(\frac{{{\text{d}}A}}{{{\text{d}}r}}\).
• 16N.2.sl.TZ0.6f: Using your answer to part (e), find the value of \(r\) which minimizes \(A\).
• 16N.2.sl.TZ0.6g: Find the value of this minimum area.
• 16N.2.sl.TZ0.6h: Find the least number of cans of water-resistant material that will coat the area in part (g).
• 16M.2.sl.TZ2.5a: Hugo is given a rectangular piece of thin cardboard, \(16\,{\text{cm}}\) by \(10\,{\text{cm}}\)....
• 16M.2.sl.TZ2.5b: (i)     State whether \(x\) can have a value of \(5\). Give a reason for your answer. (ii)  ...
• 16M.2.sl.TZ2.5c: Show that the volume, \(V\,{\text{c}}{{\text{m}}^3}\), of this tray is given...
• 16M.2.sl.TZ2.5d: Find \(\frac{{dV}}{{dx}}.\)
• 16M.2.sl.TZ2.5e: Using your answer from part (d), find the value of \(x\) that maximizes the volume of the tray.
• 16M.2.sl.TZ2.5f: Calculate the maximum volume of the tray.
• 16M.2.sl.TZ2.5g: Sketch the graph of \(V = 4{x^3} - 51{x^2} + 160x\) , for the possible values of \(x\) found...
• 17M.2.sl.TZ1.6a: Find \(g'(x)\).
• 17M.2.sl.TZ1.6b.i: Show that \(k = 6\).
• 17M.2.sl.TZ1.6b.ii: Find the equation of the tangent to the graph of \(y = g(x)\) at \(x = 2\). Give your answer in...
• 17M.2.sl.TZ1.6c: Use your answer to part (a) and the value of \(k\), to find the \(x\)-coordinates of the...
• 17M.2.sl.TZ1.6d.i: Find \(g’( - 1)\).
• 17M.2.sl.TZ1.6d.ii: Hence justify that \(g\) is decreasing at \(x =  - 1\).
• 17M.2.sl.TZ1.6e: Find the \(y\)-coordinate of the local minimum.
• 17M.2.sl.TZ2.6a: Write down the \(y\)-intercept of the graph.
• 17M.2.sl.TZ2.6b: Find \(f'(x)\).
• 17M.2.sl.TZ2.6c.i: Show that \(a = 8\).
• 17M.2.sl.TZ2.6d.i: Write down the \(x\)-coordinates of these two points;
• 17M.2.sl.TZ2.6e: Write down the range of \(f(x)\).
• 17M.2.sl.TZ2.6f: Write down the number of possible solutions to the equation \(f(x) = 5\).
• 17M.2.sl.TZ2.6g: The equation \(f(x) = m\), where \(m \in \mathbb{R}\), has four solutions. Find the possible...
• 17M.2.sl.TZ2.6c.ii: Find \(f(2)\).
• 17M.2.sl.TZ2.6d.ii: Write down the intervals where the gradient of the graph of \(y = f(x)\) is positive.
• 17N.1.sl.TZ0.11a: Find the equation of the axis of symmetry of the graph of \(y = f(x)\).
• 17N.1.sl.TZ0.11b: Write down the value of \(c\).
• 17N.1.sl.TZ0.11c: Find the value of \(a\) and of \(b\).
• 18M.2.sl.TZ1.4a: Find the value of k.
• 18M.2.sl.TZ1.4b: Using your value of k , find f ′(x).
• 18M.2.sl.TZ1.4c: Use your answer to part (b) to show that the minimum value of f(x) is −22 .
• 18M.2.sl.TZ1.4d: Write down the two values of x which satisfy f (x) = 0.
• 18M.2.sl.TZ1.4e: Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.Clearly indicate the minimum...
• 18M.1.sl.TZ2.13a: Find the cost of producing 70 shirts.
• 18M.1.sl.TZ2.13b: Find the value of s.
• 18M.1.sl.TZ2.13c: Find the number of shirts produced when the cost of production is lowest.
• 18M.2.sl.TZ2.6a: Sketch the curve for −1 < x < 3 and −2 < y < 12.
• 18M.2.sl.TZ2.6b: A teacher asks her students to make some observations about the curve. Three students...
• 18M.2.sl.TZ2.6c: Find the value of y when x = 1 .
• 18M.2.sl.TZ2.6d: Find \(\frac{{{\text{dy}}}}{{{\text{dx}}}}\).
• 18M.2.sl.TZ2.6e: Show that the stationary points of the curve are at x = 1 and x = 2.
• 18M.2.sl.TZ2.6f: Given that y = 2x3 − 9x2 + 12x + 2 = k has three solutions, find the possible values of k.
• 16N.2.sl.TZ0.6d: Show that \(A = \pi {r^2} + \frac{{1\,000\,000}}{r}\).

7.6

• 10M.2.sl.TZ2.5c: Differentiate A in terms of x.
• 10M.2.sl.TZ2.5d: Find the value of x that makes A a minimum.
• 10M.2.sl.TZ2.5e: Calculate the minimum total surface area of the dog food can.
• 12M.2.sl.TZ2.5c: Find \( \frac{{\text{d}V}}{{\text{d}x}}\).
• 12M.2.sl.TZ2.5d: Find the value of x for which V is a maximum.
• 12M.2.sl.TZ2.5e: Find the maximum volume of the container.
• 12M.2.sl.TZ2.5f: Find the length and height of the container for which the volume is a maximum.
• SPM.1.sl.TZ0.15c: Use your answer to part (b) to find the selling price of each machine in order to maximize...
• SPM.2.sl.TZ0.6f: (i)     Find the value of \(r\) that minimizes the total external surface area of the wastepaper...
• SPM.2.sl.TZ0.6g: Determine whether Merryn’s design is an improvement upon Nadia’s. Give a reason.
• 08M.2.sl.TZ1.5ii.e: (i)     Hence find the value of \(x\) and of \(y\) required to make the volume of the box a...
• 08M.2.sl.TZ2.4ii.c: Calculate the minimum cost per person.
• 14M.2.sl.TZ2.5f: The parcel is tied up using a length of string that fits exactly around the parcel, as shown in...
• 14M.2.sl.TZ2.5g: The parcel is tied up using a length of string that fits exactly around the parcel, as shown in...
• 14M.2.sl.TZ2.5h: The parcel is tied up using a length of string that fits exactly around the parcel, as shown in...
• 14M.2.sl.TZ1.6e: The lobster trap is designed so that the length of steel used in its frame is a minimum. Show...
• 14M.2.sl.TZ1.6f: The lobster trap is designed so that the length of steel used in its frame is a...
• 14M.2.sl.TZ1.6g: The lobster trap is designed so that the length of steel used in its frame is a...
• 14N.2.sl.TZ0.3f: A company designs cone-shaped tents to resemble the traditional tepees. These cone-shaped tents...
• 15M.1.sl.TZ1.15b: Find the value of \(x\) that makes the volume a maximum.
• 16M.1.sl.TZ1.15a: A company sells fruit juices in cylindrical cans, each of which has a volume of...
• 16M.1.sl.TZ1.15b: Calculate the value of \(r\) that minimizes the surface area of a can.
• 16N.2.sl.TZ0.6a: Write down a formula for \(A\), the surface area to be coated.
• 16N.2.sl.TZ0.6b: Express this volume in \({\text{c}}{{\text{m}}^3}\).
• 16N.2.sl.TZ0.6c: Write down, in terms of \(r\) and \(h\), an equation for the volume of this water container.
• 16N.2.sl.TZ0.6e: Find \(\frac{{{\text{d}}A}}{{{\text{d}}r}}\).
• 16N.2.sl.TZ0.6f: Using your answer to part (e), find the value of \(r\) which minimizes \(A\).
• 16N.2.sl.TZ0.6g: Find the value of this minimum area.
• 16N.2.sl.TZ0.6h: Find the least number of cans of water-resistant material that will coat the area in part (g).
• 16M.2.sl.TZ2.5a: Hugo is given a rectangular piece of thin cardboard, \(16\,{\text{cm}}\) by \(10\,{\text{cm}}\)....
• 16M.2.sl.TZ2.5b: (i)     State whether \(x\) can have a value of \(5\). Give a reason for your answer. (ii)  ...
• 16M.2.sl.TZ2.5c: Show that the volume, \(V\,{\text{c}}{{\text{m}}^3}\), of this tray is given...
• 16M.2.sl.TZ2.5d: Find \(\frac{{dV}}{{dx}}.\)
• 16M.2.sl.TZ2.5e: Using your answer from part (d), find the value of \(x\) that maximizes the volume of the tray.
• 16M.2.sl.TZ2.5f: Calculate the maximum volume of the tray.
• 16M.2.sl.TZ2.5g: Sketch the graph of \(V = 4{x^3} - 51{x^2} + 160x\) , for the possible values of \(x\) found...
• 17N.1.sl.TZ0.15a: Write down how many kilograms of cheese Maria sells in one week if the price of a kilogram of...
• 17N.1.sl.TZ0.15b: Find how much Maria earns in one week, from selling cheese, if the price of a kilogram of cheese...
• 17N.1.sl.TZ0.15c: Write down an expression for \(W\) in terms of \(p\).
• 17N.1.sl.TZ0.15d: Find the price, \(p\), that will give Maria the highest weekly profit.
• 18M.2.sl.TZ1.6a: Write down the height of the cylinder.
• 18M.2.sl.TZ1.6b: Find the total volume of the trash can.
• 18M.2.sl.TZ1.6c: Find the height of the cylinder, h , of the new trash can, in terms of r.
• 18M.2.sl.TZ1.6d: Show that the volume, V cm3 , of the new trash can is given by \(V = 110\pi {r^3}\).
• 18M.2.sl.TZ1.6e: Using your graphic display calculator, find the value of r which maximizes the value of V.
• 18M.2.sl.TZ1.6f: The designer claims that the new trash can has a capacity that is at least 40% greater than the...
• 18M.1.sl.TZ2.13a: Find the cost of producing 70 shirts.
• 18M.1.sl.TZ2.13b: Find the value of s.
• 18M.1.sl.TZ2.13c: Find the number of shirts produced when the cost of production is lowest.
• 16N.2.sl.TZ0.6d: Show that \(A = \pi {r^2} + \frac{{1\,000\,000}}{r}\).