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Practice sheet · Calculus Path · Lens 1: Math

Derivatives & integrals

Practice for The Math of Calculus: rates, limits, derivatives, tangents, max and min, Riemann sums and integrals (pp.8–20), plus Math 31 Level Ups (pp.22–26).
Warm-upCoreLevel Up
Name Date Time I spent
Remember

Power rule: d/dx xⁿ = nxⁿ⁻¹. The derivative f′(a) is the slope of the tangent at x = a. At a max or min, f′ = 0; f″ > 0 means a valley, f″ < 0 a hill.

Integral: area under the curve. If F′ = f, then ∫ from a to b of f dx = F(b) − F(a). Chain rule: d/dx f(g(x)) = f′(g(x)) · g′(x).

Warm-up

One step

Rates, limits and the power rule.
1

Average rate

A ball rolls s = t² metres after t seconds. What is its average speed from t = 1 to t = 3?

2

Power rule

Differentiate x⁵ and 3x².

3

Slope at a point

For f(x) = x², find f′(3).

4

A limit

Find the limit of (x² − 4) ÷ (x − 2) as x → 2. (Factor first.)

Core

Two or three steps

Show the derivative or the sum.
5

Tangent line

Find the equation of the tangent to y = x³ at x = 1.

6

Falling ball

A dropped ball falls s = 4.9t² metres. Find its velocity v = s′(t). How fast is it falling at t = 2 s?

Calculus Practice · Calculus Path · Unizon1/2
Calculus Path · Unizon · PracticeDerivatives & integrals
7

Lowest point

Find the minimum of f(x) = x² − 6x + 10. Where is it, and what is the value?

8

Hill and valley

For f(x) = x³ − 3x, find where f′(x) = 0. Use f″ to say which is a max and which a min, and give their values.

9

Rectangles

Estimate the area under y = x² from 0 to 2 with 4 right-hand rectangles of width 0.5. Compare with the exact area, 8/3.

10

Integrals

Evaluate ∫ from 0 to 3 of 2x dx, and ∫ from 1 to 4 of 3x² dx.

Level Up

Math 31 Level Ups

Chain and product rules, optimizing, related rates (pp.22–26).
11

Chain rule

Differentiate (2x + 1)⁴. What is the slope at x = 0?

12

Product rule

Differentiate x³(x − 2) with the product rule. Check by expanding first. Find the slope at x = 2.

13

Best pen

40 m of fence makes a rectangle. Write its area as A(x) = x(20 − x), then use A′(x) = 0 to find the biggest area.

14

Growing ripple

A circle’s radius grows at 2 cm/s. How fast is its area growing when r = 5 cm? Use dA/dt = 2πr · dr/dt.

Make your own problem

Your turn to be the calculus teacher

  1. Pick a function or a moving object from the book.
  2. Write a question about its rate, its slope, its best value or an area.
  3. Solve it, then check with a table of values or a quick sketch.

My problem, and my worked answer:

Calculus Practice · Calculus Path · Unizon2/2
Calculus Path · Unizon · PracticeDerivatives & integrals

Try every problem first, then check here. Wrong answers are useful: find the step where your work and the key split apart.

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Answer key

Worked solutions. Level Ups match Alberta Math 31 topics.

Warm-up
1
4 m/s (3² − 1²) ÷ (3 − 1) = 8 ÷ 2 = 4.
2
5x⁴ · 6x Bring the power down and subtract one: 5x⁴. 3 × 2x = 6x.
3
6 f′(x) = 2x, so f′(3) = 6.
4
4 (x − 2)(x + 2) ÷ (x − 2) = x + 2 for x ≠ 2. As x → 2, it → 4.
Core
5
y = 3x − 2 y′ = 3x² = 3 at x = 1. The point is (1, 1): y − 1 = 3(x − 1).
6
v = 9.8t · 19.6 m/s s′(t) = 2 × 4.9t = 9.8t. At t = 2: 19.6 m/s (ignoring air).
7
minimum 1 at x = 3 f′(x) = 2x − 6 = 0 → x = 3. f(3) = 9 − 18 + 10 = 1. f″ = 2 > 0: a valley.
8
max 2 at x = −1 · min −2 at x = 1 f′ = 3x² − 3 = 0 → x = ±1. f″ = 6x: −6 at −1 (hill), 6 at 1 (valley). f(−1) = 2, f(1) = −2.
9
3.75 · exact ≈ 2.67 0.5 × (0.25 + 1 + 2.25 + 4) = 3.75. Right-hand rectangles overshoot a rising curve. More, thinner rectangles get closer.
10
9 · 63 x² from 0 to 3: 9 − 0 = 9. x³ from 1 to 4: 64 − 1 = 63.
Level Up
11
8(2x + 1)³ · 8 Outside: 4(2x + 1)³. Inside: 2. So 8(2x + 1)³; at x = 0, 8.
12
4x³ − 6x² · 8 3x²(x − 2) + x³ · 1 = 4x³ − 6x². Expanding: x⁴ − 2x³ → 4x³ − 6x². At 2: 32 − 24 = 8.
13
10 m × 10 m · 100 m² Two sides add to 20, so A = x(20 − x). A′ = 20 − 2x = 0 → x = 10. A = 100 m², a square.
14
20π ≈ 62.8 cm²/s 2π × 5 × 2 = 20π. The area grows faster as the circle gets bigger.
Make your own: write f′ before you substitute, check a slope with a tiny step (f(a + 0.001) − f(a)) ÷ 0.001, and sketch the curve.
Calculus Practice · Calculus Path · UnizonAnswer key