MATH1013 Calculus I, 2013-14 Spring

Tutorial Worksheet 6: Derivatives (T1A)

Name: ID No.: Tutorial Section:

Complete at least TWO questions from the following questions.

(Solution of this worksheet will be available at the course website the week after.)

1. (Demonstration) (page 138, Q. 25) Compute the derivative of√x+ 3 from the definition of a deriva-

tive, and then use the result to find the equation of tangent to√x+ 3 at x = 1.

2. (Demonstration) (page 139, Q. 51) Plot a graph of f ′(x) against the given graph of f(x) in thefollowing figure:

3. (Demonstration) (page 139, Q. 61) Use the points A, B, C, D, E in the following graphs of to answer

• At which points is the gradient of the curve negative?

• At which points is the gradient of the curve positive?

• Using A− E to list the gradients in decreasing order.

4. (Demonstration) Determine if the function f(x) =

x sin1

x, if x 6= 0;

0, if x = 0.is continuous and if it has a

derivative at x = 0.

5. (Demonstration) (page 188, Q. 77) Assume f and g are differentiable on their domains with h(x) =f(g(x)). Suppose the equation of the tangent line of g at the point (4, 7) is y = 3x− 5 and tangent lineof f at the point (7, 9) is y = −2x+ 23. (i) Calculate h(4) and h′(4), and (ii) to determine the tangentline to h at x = 4.

6. (Demonstration) (page 169, Q. 81) Identify which f and x = a so that its derivative f ′(a) is given by

the limit: limh→0

sin(π6 + h)− 12

h. Also evaluate this f ′(a).

7. (Demonstration) (p. 187, Q. 44, 49) Given that (ex)′ = ex. Differentiate sin(sin(ex)) and tan(e√3x).

8. (Demonstration) (p.168, Q. 75) Prove that (cosx)′ = − sinx,

9. (Class work) (page 138, Q. 24) Compute the derivative of√x− 1 from the definition of a derivative,

and then use the result to find the equation of tangent to√x− 1 at x = 2.

1

Answer

10. (Class work) (page 139, Q. 51) Plot a graph of f ′(x) against the given graph of f(x) in the followingfigure:

Answer

11. (Class work) (page 139, Q. 49) Match the functions a − d on the first set of figures to the second setA−D which are their derivatives:

,

Answer

12. (Class work) (p. 140, Q. 64) Find f from the given f ′(x) =

1, if x < 0;

0, if 0 < x < 1;

−1, if x > 1.

Is more than one f

possible?

Answer

13. (Class work) Compute the derivative of1

(x+ 1)2from the definition of derivative.

2

Answer

14. (Class work) Determine if the function f(x) =

x2 sin1

x, if x 6= 0;

0, if x = 0.is continuous and if it has a

derivative at x = 0.

Answer

15. (Class work) (page 188, Q. 78) Assume f is a differentiable function whose graph passes through thepoint (1, 4). Suppose g(x) = f(x2) and the tangent line of f at the point (1, 4) is y = 3x+1. Determineeach of (a) g(1), (b) g′(x), (c) g′(1), (d) an equation of the tangent line to graph of g when x = 1.

Answer

16. (Class work) (page 188, Q. 80) Suppose f is differentiable on [−2, 2] with f ′(0) = 3 and f ′(1) = 5.Let g(x) = f(sinx). Evaluate the (a) g′(0), (b) g′(π/2), (c) g′(π).

Answer

17. (Class work) (page 169, Q. 81) Identify which f and x = a so that its derivative f ′(a) is given by the

limit: limh→0

cos(π6 + h)−√32

h. Also evaluate this f ′(a).

Answer

3

18. (Class work) (p.187, Q. 46, 58) Given that (ex)′ = ex. Differentiate sin2(e3x+1) and tan(xex).

Answer

19. (Class work) (p.189, Q. 86) Given that (ex)′ = ex. Show that y(t) = e−t(sin 2t − 2 cos 2t) satisfiesy′′(t) + 2y′(t) + 5y(t) = 0.

Answer

20. (Class work) (p.189, Q. 93) Using the product and chain rules to derive the quotient rule:(fg

)′=f ′(x)g(x)− f(x)g′(x)

g(x)2.

Answer

4