A curve has equation y = x2 + 6x - 2. Find the gradient function dy/dx.
(Total for Question 1 is 1 mark)
2
A curve has equation y = x3 - 7x. Find dy/dx.
(Total for Question 2 is 1 mark)
3
A curve has equation y = x2 - 4x + 1. Calculate the gradient of the curve at the point where x = 5.
(Total for Question 3 is 2 marks)
4
A curve has equation y = x3 - 3x2. Calculate the gradient of the curve at the point where x = 1.
(Total for Question 4 is 2 marks)
5
A curve has equation y = 2x2 - 5x + 1. Find the value of x at which the gradient of the curve equals 7.
(Total for Question 5 is 2 marks)
6
A curve has equation y = x2 + 2x - 8. Find the coordinates of the point on the curve where the gradient is zero.
(Total for Question 6 is 2 marks)
7
State, without further calculation, whether the curve y = x2 - 12x + 40 has a minimum or a maximum turning point.
(Total for Question 7 is 1 mark)
8
State, without further calculation, whether the curve y = -x2 + 4x + 1 has a minimum or a maximum turning point.
(Total for Question 8 is 1 mark)
9
A curve has equation y = x2 - 10x + 21. Find the x-coordinate of the turning point of the curve.
(Total for Question 9 is 2 marks)
10
A curve has equation y = x3 - 12x. Find the x-coordinates of the stationary points of the curve.
(Total for Question 10 is 2 marks)
11
A curve has equation y = 3x2 - 2. Find the second derivative d2y/dx2.
(Total for Question 11 is 1 mark)
12
A curve has equation y = x3 - 6x2 + 2. Find the second derivative d2y/dx2.
(Total for Question 12 is 2 marks)
13
A curve has equation y = x2 - 8x + 19. Find the coordinates of the turning point of the curve and state whether it is a minimum or a maximum.
(Total for Question 13 is 3 marks)
14
A curve has equation y = x3 - 3x2 - 9x + 10. Show that the gradient of the curve at the point where x = -2 is 15.
(Total for Question 14 is 3 marks)
15
A curve has equation y = x2 - 6x + 5. Find the equation of the tangent to the curve at the point where x = 1.
(Total for Question 15 is 4 marks)
16
A curve has equation y = x3 - 27x + 10. Find the coordinates of the stationary points of the curve and use the second derivative to determine the nature of each.
(Total for Question 16 is 4 marks)
17
A curve has equation y = 2x3 + 3x2 - 36x + 5. Find the range of values of x for which the curve is a decreasing function.
(Total for Question 17 is 3 marks)
18
A firework rocket is launched vertically. Its height above the ground, s metres, t seconds after launch is modelled by s = 24t - 4t2, for 0 ≤ t ≤ 6.
(a)Find ds/dt.(2)
(b)Find the maximum height reached by the rocket, and the time at which it occurs.(2)
(Total for Question 18 is 4 marks)
Mark scheme · C2D Applications of Differentiation: Gradients, Tangents and Turning Points: Fluency and Exam Drill
Question 1
B1 dy/dx = 2x + 6 cao
Answer: dy/dx = 2x + 6
Question 2
B1 dy/dx = 3x2 - 7 cao
Answer: dy/dx = 3x2 - 7
Question 3
M1 differentiates to obtain dy/dx = 2x - 4 and substitutes x = 5
A1 gradient = 6 cao
Answer: gradient = 6
Question 4
M1 differentiates to obtain dy/dx = 3x2 - 6x and substitutes x = 1
A1 gradient = -3 cao
Answer: gradient = -3
Question 5
M1 differentiates and sets 4x - 5 = 7
A1 x = 3 cao
Answer: x = 3
Question 6
M1 differentiates and sets 2x + 2 = 0 to find x = -1
A1 point (-1, -9) cao
Answer: (-1, -9)
Question 7
B1 minimum, since the coefficient of x2 is positive
Answer: Minimum
Question 8
B1 maximum, since the coefficient of x2 is negative
Answer: Maximum
Question 9
M1 differentiates to obtain dy/dx = 2x - 10 and sets equal to 0
A1 x = 5 cao
Answer: x = 5
Question 10
M1 differentiates to obtain dy/dx = 3x2 - 12 and sets equal to 0
A1 x = 2 and x = -2 (both values) cao
Answer: x = 2 or x = -2
Question 11
B1 d2y/dx2 = 6 cao
Answer: d2y/dx2 = 6
Question 12
M1 differentiates once to obtain dy/dx = 3x2 - 12x
A1 d2y/dx2 = 6x - 12 cao
Answer: d2y/dx2 = 6x - 12
Question 13
M1 differentiates and sets 2x - 8 = 0 to find x = 4
A1 y = 3, so the point is (4, 3)
B1 minimum, since the coefficient of x2 is positive (or d2y/dx2 = 2 > 0)