1.
Differentiate y = 5x⁴ − 3x² + 7. [2]
JD SCIENCE
Pure — differentiation and integration
Answer all questions. Show working. Answers are in a separate file on the Worksheets page — do not look at them until you have finished.
2.
Differentiate y = (2x + 1)³. [3]
3.
Differentiate y = x² sin x. [3]
4.
Differentiate y = (x² + 1)/(x − 1). [4]
5.
Find dy/dx if x = 2t + 1, y = t². [3]
6.
Find the gradient of y = x³ − 3x at x = 2. [2]
7.
Find the equation of the tangent to y = x² at (3, 9). [3]
8.
Find the equation of the normal to y = x² at (3, 9). [3]
9.
Find the stationary points of y = x³ − 12x + 1 and determine their nature. [5]
10.
A rectangle has perimeter 20 cm. Show that the area A = 10x − x² and find the maximum area. [5]
11.
Integrate 6x² − 4x + 3. [2]
12.
Find ∫ (2x + 1)⁴ dx. [3]
13.
Evaluate ∫₀² (3x² + 1) dx. [3]
14.
Find the area between y = x², the x-axis, x = 1 and x = 3. [3]
15.
Find the area between y = x and y = x² from 0 to 1. [3]
16.
Use the trapezium rule with 4 strips to estimate ∫₀⁴ (x²) dx. Comment on whether this is an over- or underestimate. [4]
17.
If dy/dx = 6x² and y = 5 when x = 1, find y. [3]
18.
Differentiate y = e^{3x} and y = ln(5x). [3]
19.
Differentiate y = e^{x²}. [2]
20.
Find ∫ 1/(2x) dx, x > 0. [2]
21.
Implicit differentiation: x² + y² = 25. Find dy/dx at (3, 4). [3]
22.
Related rates: a sphere has V = (4/3)πr³. If dr/dt = 2, find dV/dt when r = 3. [4]
23.
Find the second derivative of y = sin 2x. [2]
24.
Explain how to decide whether a stationary point is a max, min or point of inflection using the second derivative. [3]
25.
Integration by substitution: ∫ 2x(x² + 1)³ dx. [3]
26.
Find the average value of y = 3x² on [0, 2]. [3]
27.
A curve is y = 1/x. Find the volume generated when the region from x = 1 to x = 2 is rotated about the x-axis. Leave π in the answer if this is on the spec; otherwise find the area under the curve. [3]
28.
State the chain, product and quotient rules in Leibniz or function notation. [3]
29.
A OCR question gives displacement s = t³ − 6t². Find the velocity and acceleration at t = 1. [3]
30.
Solve dy/dx = 2y, y(0) = 3, by separating variables. [4]
31.
Why do you add + c for an indefinite integral but not for a definite integral? [2]
32.
Find the x-coordinate of the point on y = x² + 1 where the tangent is parallel to y = 4x. [3]