Calculus 1 Practice Test

Name: ______________________   Date: __________   Score: ____ / 30

30 questions. Calculator allowed. Show your working. steplyclass.com/calculus-1/practice-test/

1. 1

   Find the derivative of sin⁻¹(x).

   Answer: ____________________

2. 2

   Find the derivative of tan⁻¹(3x + 2).

   Answer: ____________________

3. 3

   Find the derivative of 3^2x.

   Answer: ____________________

4. 4

   Find the derivative of 4log_3(2x + 1).

   Answer: ____________________

5. 5

   Find the derivative of y = (x + 1)² (x − 3)⁴ using logarithms.

   Answer: ____________________

6. 6

   Find the derivative of y = (x + 1)² (x + 3)³/(x + 2) using logarithms.

   Answer: ____________________

7. 7

   Linear approximation of √x at x = 4

   Answer: ____________________

8. 8

   Estimate sin(0.05) with a linear approximation

   Answer: ____________________

9. 9

   Find dy for y = x² when x = 3 and dx = 0.1
   1. A) 0.06
   2. B) 6
   3. C) 0.6

10. 10

    The cube has side 10 cm, measured with a possible error of 0.1 cm. Estimate the percentage error in the volume.

    Answer: ____________________

11. 11

    Absolute max and min of f(x) = x³ − 3x on [−2, 2]
    1. A) 2 and x = 1
    2. B) −2 and x = 1
    3. C) 2 and x = −1
    4. D) −2

12. 12

    Absolute maximum and minimum of f(x) = −x³ + 3x² on [−1, 3]
    1. A) 0 and x = −3
    2. B) 0 and x = 3
    3. C) x = 3
    4. D) 0

13. 13

    Solve Rolle's theorem for f(x) = x² − 4x on [0, 4]
    1. A) c = 4
    2. B) c = 6
    3. C) c = 2
    4. D) c = 0

14. 14

    Solve Rolle's theorem for f(x) = x³ − x on [0, 1]

    Answer: ____________________

15. 15

    Concavity and inflection points of f(x) = x³ − 3x²

    Answer: ____________________

16. 16

    Inflection points of f(x) = x⁴ + 4x³

    Answer: ____________________

17. 17

    Two positive numbers multiply to 64. Find the smallest possible sum.
    1. A) 16
    2. B) 1
    3. C) 64

18. 18

    An open box with a square base must hold 108 cm³. Find the least material needed.

    Answer: ____________________

19. 19

    Find the limit as x -> 0 of sin(x)/x.
    1. A) 0
    2. B) −1
    3. C) 1

20. 20

    Find the limit as x -> 0 of (1 − cos(2x))/x^2.

    Answer: ____________________

21. 21

    Use Newton's method on x² − 2 = 0 starting at x = 1
    1. A) root ≈ 14.1667
    2. B) root ≈ −1
    3. C) root ≈ 1.41667
    4. D) root ≈ 2

22. 22

    Estimate √3 with Newton's method from x = 2, 3 steps

    Answer: ____________________

23. 23

    f′(x) = 2x, f(0) = 4, find f(x)

    Answer: ____________________

24. 24

    dy/dx = 6x² − 6x + 5, y = 1 when x = 0, find y

    Answer: ____________________

25. 25

    Find d/dx ∫ from 1 to x of (t² + 1) dt

    Answer: ____________________

26. 26

    Find d/dx ∫ from 0 to 2x of e^t dt

    Answer: ____________________

27. 27

    Find the average value of f(x) = 2x + 1 on [0, 4].

    Answer: ____________________

28. 28

    Average value of f(x) = x² on [0, 3], and find c

    Answer: ____________________

29. 29

    Water flows into a tank at r(t) = 4t + 5 litres per minute. How much flows in from t = 0 to t = 4?

    Answer: ____________________

30. 30

    V(t) = t² − 2t ms from t = 1 to t = 4. Find the displacement and the total distance.

    Answer: ____________________