DP Mathematical Studies Questionbank
7.5
Description
[N/A]Directly related questions
- 16N.2.sl.TZ0.6d: Show that \(A = \pi {r^2} + \frac{{1\,000\,000}}{r}\).
- 18M.2.sl.TZ2.6f: Given that y = 2x3 − 9x2 + 12x + 2 = k has three solutions, find the possible values of k.
- 18M.2.sl.TZ2.6e: Show that the stationary points of the curve are at x = 1 and x = 2.
- 17N.1.sl.TZ0.11c: Find the value of \(a\) and of \(b\).
- 17N.1.sl.TZ0.11b: Write down the value of \(c\).
- 17N.1.sl.TZ0.11a: Find the equation of the axis of symmetry of the graph of \(y = f(x)\).
- 17M.2.sl.TZ2.6g: The equation \(f(x) = m\), where \(m \in \mathbb{R}\), has four solutions. Find the possible...
- 17M.2.sl.TZ2.6f: Write down the number of possible solutions to the equation \(f(x) = 5\).
- 17M.2.sl.TZ2.6e: Write down the range of \(f(x)\).
- 17M.2.sl.TZ2.6d.ii: Write down the intervals where the gradient of the graph of \(y = f(x)\) is positive.
- 17M.2.sl.TZ2.6d.i: Write down the \(x\)-coordinates of these two points;
- 17M.2.sl.TZ2.6c.ii: Find \(f(2)\).
- 17M.2.sl.TZ2.6c.i: Show that \(a = 8\).
- 17M.2.sl.TZ2.6b: Find \(f'(x)\).
- 17M.2.sl.TZ2.6a: Write down the \(y\)-intercept of the graph.
- 17M.2.sl.TZ1.6e: Find the \(y\)-coordinate of the local minimum.
- 17M.2.sl.TZ1.6d.ii: Hence justify that \(g\) is decreasing at \(x = - 1\).
- 17M.2.sl.TZ1.6d.i: Find \(g’( - 1)\).
- 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.TZ1.6b.ii: Find the equation of the tangent to the graph of \(y = g(x)\) at \(x = 2\). Give your answer in...
- 17M.2.sl.TZ1.6b.i: Show that \(k = 6\).
- 17M.2.sl.TZ1.6a: Find \(g'(x)\).
- 16M.2.sl.TZ2.5g: Sketch the graph of \(V = 4{x^3} - 51{x^2} + 160x\) , for the possible values of \(x\) found...
- 16M.2.sl.TZ2.5f: Calculate the maximum volume of the tray.
- 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.TZ2.5d: Find \(\frac{{dV}}{{dx}}.\)
- 16M.2.sl.TZ2.5c: Show that the volume, \(V\,{\text{c}}{{\text{m}}^3}\), of this tray is given...
- 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.TZ2.5a: Hugo is given a rectangular piece of thin cardboard, \(16\,{\text{cm}}\) by \(10\,{\text{cm}}\)....
- 16M.2.sl.TZ1.3f: The nearest coastguard can see the flare when its height is more than \(40\) metres above sea...
- 16M.2.sl.TZ1.3e: i) Show that the flare reached its maximum height \(40\) seconds after being fired. ii) ...
- 16M.2.sl.TZ1.3d: Find \(h'\,(t)\,.\)
- 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.TZ1.3b: Find the height of the flare \(15\) seconds after it was fired.
- 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...
- 16N.2.sl.TZ0.6h: Find the least number of cans of water-resistant material that will coat the area in part (g).
- 16N.2.sl.TZ0.6g: Find the value of this minimum area.
- 16N.2.sl.TZ0.6f: Using your answer to part (e), find the value of \(r\) which minimizes \(A\).
- 16N.2.sl.TZ0.6e: Find \(\frac{{{\text{d}}A}}{{{\text{d}}r}}\).
- 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.TZ0.6b: Express this volume in \({\text{c}}{{\text{m}}^3}\).
- 16N.2.sl.TZ0.6a: Write down a formula for \(A\), the surface area to be coated.
- 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.2.sl.TZ0.5g: Find the coordinates of the vertex of P and state the gradient of the curve at this point.
- 07N.1.sl.TZ0.15b: The function \(f (x)\) has a local maximum at the point where \(x = −1\). Find the value of a.
- 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...
- 18M.2.sl.TZ2.6d: Find \(\frac{{{\text{dy}}}}{{{\text{dx}}}}\).
- 18M.2.sl.TZ2.6c: Find the value of y when x = 1 .
- 18M.2.sl.TZ2.6b: A teacher asks her students to make some observations about the curve. Three students...
- 18M.2.sl.TZ2.6a: Sketch the curve for −1 < x < 3 and −2 < y < 12.
- 18M.1.sl.TZ2.13c: Find the number of shirts produced when the cost of production is lowest.
- 18M.1.sl.TZ2.13b: Find the value of s.
- 18M.1.sl.TZ2.13a: Find the cost of producing 70 shirts.
- 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.2.sl.TZ1.4d: Write down the two values of x which satisfy f (x) = 0.
- 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.TZ1.4a: Find the value of k.
- 18M.2.sl.TZ1.4b: Using your value of k , find f ′(x).
- 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.TZ1.5b: The graph of \(y = f(x)\) has a local minimum point at \(x = 4\). Show that \(k = 3\).
- 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.TZ1.5d: The graph of \(y = f(x)\) has a local minimum point at \(x = 4\). Find \(f'(2)\)
Sub sections and their related questions
Values of x where the gradient of a curve is zero.
- 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...
- 07M.2.sl.TZ0.3i.c: Using your graphic display calculator or otherwise, write down the coordinates of any point where...
- 07N.2.sl.TZ0.5g: Find the coordinates of the vertex of P and state the gradient of the curve at this point.
- 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...
- 16N.2.sl.TZ0.6a: Write down a formula for \(A\), the surface area to be coated.
- 17M.2.sl.TZ1.6b.i: Show that \(k = 6\).
- 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.
Solution of \(f'\left( x \right) = 0\).
- 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...
- 09N.1.sl.TZ0.6c: Find the value of \(x\) for which \(f'(x) = 0\).
- 09M.2.sl.TZ1.5d: Solve the equation \(f ′(x) = 0\).
- 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...
- 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.TZ0.15b: Find the number of machines that should be made and sold each month to maximize \(P(x)\) .
- SPM.2.sl.TZ0.6f: (i) Find the value of \(r\) that minimizes the total external surface area of the wastepaper...
- 14M.1.sl.TZ1.15b: The curve has a local minimum at the point where \(x = 2\). Find the value of \(k\).
- 15M.1.sl.TZ1.15b: Find the value of \(x\) that makes the volume a maximum.
- 15M.2.sl.TZ1.5b: The graph of \(y = f(x)\) has a local minimum point at \(x = 4\). Show that \(k = 3\).
- 16N.2.sl.TZ0.6a: Write down a formula for \(A\), the surface area to be coated.
- 17M.2.sl.TZ1.6b.i: Show that \(k = 6\).
- 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.
Stationary points.
- 16N.2.sl.TZ0.6a: Write down a formula for \(A\), the surface area to be coated.
- 17M.2.sl.TZ1.6b.i: Show that \(k = 6\).
- 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.
Local maximum and minimum points.
- 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.2.sl.TZ0.5A, f: Use Differential Calculus to verify that your answer to (e) is correct.
- 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;
- 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.
- 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.
- 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.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...
- 16N.2.sl.TZ0.6a: Write down a formula for \(A\), the surface area to be coated.
- 17M.2.sl.TZ1.6b.i: Show that \(k = 6\).
- 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.
