• 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 $A = \pi {r^2} + \frac{{1\,000\,000}}{r}$.