Tutorial 7 (Complete Solutions)

MATHEMATICS 101

TUTORIAL 7 (COMPLETE SOLUTIONS)

Textbook: Calculus (Stewart)

MODULE 3 – DIFFERENTIAL CALCULUS

Exercise 3 Review (page 215)

Review of Differentiation

Question 15

Find if .

Solution

Question 17

Find if .

Solution

Question 21 (D)

Find if .

Solution

Question 45

Find the equation of the tangent to the curve at the point .

Solution

The gradient function .

At .

Equation of tangent: .

Question 46 (D)

Find the equation of the tangent to the curve at the point .

Solution

The gradient function .

At .

Equation of tangent: .

Exercise 3.9 (page 202)

Related rates

Question 3

If and , find when x = 2.

Solution

Question 5

If , and , find when x = 5 and y = 12.

Solution

When x = 5 and y = 12, .

Therefore, .

Question 6

A particle moves along the curve . As it reaches the point (2, 3), the y-coordinate is increasing at a rate of 4 cm/s. How fast is the x-coordinate of the point changing at that instant?

Solution

Therefore, cm/s.

Exercise 3.7 (page 188)

Implicit differentiation

Question 1

Given that ,

(a) Find by implicit differentiation.

(b) Solve the equation explicitly for y and differentiate to get in terms of x.

(c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution to part (a).

Solution

(a)

Thus, .

(b)

Thus, .

Question 3

Repeat Question 1 with the function .

Solution

(a)

Thus, .

(b) .

Thus, .

Question 11 (D)

Find by implicit differentiation given that .

Solution

Question 26

Use implicit differentiation to find an equation of the tangent line to the curve at the point (1, 2).

Solution

At (1, 2), the gradient of the curve is .

Therefore, the equation of tangent at (1, 2) is

Exercise 3.8 (page 195)

Higher-order derivatives

Question 15

Find the first and second derivatives of the function .

Solution

Question 29

Find by implicit differentiation if .

Solution

Thus, .

Question 57

Given that the function g is a twice-differentiable function, findin terms of g, and given that .

Solution

Exercise 3.10 (page 211)

Differentials

Question 15

Find the differential of the function .

Solution

Question 17

Find the differential of the function .

Solution

Question 21

Given that , find the differential dy and evaluate dy for x = –2 and dx = 0.1.

Solution

Question 27 (D)

Given that , compute and dy for x = 1 and dx = = 0.5. (Optional: then sketch a diagram like Fig. 6 showing the line segments with lengths dx, dy and .)

Solution

Question 39

The edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum possible error, relative error and percentage error in computing (a) the volume of the cube and (b) the surface area of the cube.

Solution

Let x be the length of one edge of the cube. Therefore, cm.

Volume = cm3.

Relative error in volume = .

Percentage error in volume = 1%.

Surface area = cm2.

Relative error in volume = .

Percentage error in volume = %.

MODULE 4 – INTEGRAL CALCULUS

Exercise 5.1 (page 325)

Review of basic principles of integration

Question 17

Use definition 2 (page 320) to find an expression for the area under the graph of , , as a limit. Do not evaluate the limit.

Solution

Area =.

Question 19

Use definition 2 (page 320) to find an expression for the area under the graph of , , as a limit. Do not evaluate the limit.

Solution

Area =.

Question 21 (D)

Determine a region whose area is equal to .

Solution

Exercise 4.10 (page 305)

Antiderivatives

Question 1

Find the most general antiderivative of . (Check your answer by differentiation.)

Solution

Question 3

Find the most general antiderivative of . (Check your answer by differentiation.)

Solution

Question 7

Find the most general antiderivative of . (Check your answer by differentiation.)

Solution

Question 11 (D)

Find the most general antiderivative of . (Check your answer by differentiation.)

Solution

Question 23

Find f given that and .

Solution

Substituting the given condition, we have .

Therefore, .

Question 27

Find f given that and for .

Solution

Substituting the given condition, we have .

Therefore, .

Hence, .

Question 37

Given that the graph of f passes through the point (1, 6) and that the slope of its tangent line at is , find .

Solution

Given that and that . Thus, .

Substituting the given condition, we have .

Therefore, and .

Question 43 (D)

The graph of is shown in Fig. 1 below. Sketch the graph of f if f is continuous and .

Fig. 1

Solution

Note that for , for , and for . We suspect that is a piecewise function since, despite being continuous, it is not differentiable at x = 1 and x = 2. Clearly, these two values of x must be the endpoints of these sub-functions (recall that a function is not differentiable at its endpoints).

For , . Substituting the condition , we have C = –1 so that . Furthermore, for , and for , . Note that and need not be determined since the sub-functions have to meet at their endpoints in order to make f continuous.

The graph of f is shown in blue in Fig. 1 above.

Exercise 5.2 (page 337)

Definite integration

Question 17

Express as a definite integral on .

Solution

Question 35

Evaluate by interpreting it in terms of areas.

Solution

(The area bounded by the curve and the x-axis).

Question 37

Evaluate by interpreting it in terms of areas.

Solution

Let . Using the trigonometric substitution , we have . Changing limits from x to : when x = 0, , when x = –3, .

Therefore, .

, using the double-angle formula.

Question 47 (D)

Already set in previous tutorial!

Question 54

Use the properties of integrals to verify the inequality

without evaluating the integrals.

Solution

By the Mean Value Theorem for Integrals,

for.

Integrating, we have .

On simplification, we obtain .

Exercise 5.3 (page 348)

Fundamental Theorem

Question 5

Already set in previous tutorial!

Question 7

Use Part I of the Fundamental Theorem of Calculus to find the derivative of .

Solution

Question 11 (D)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of .

Hint: .

Solution

Question 13 (D)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of .

Solution

Question 44

Find the derivative of the function .

Solution

Exercise 5.4 (page 356)

Fundamental Theorem

Question 1

Verify by differentiation that the formula is correct.

Solution

Question 3

Verify by differentiation that the formula is correct.

Solution

Question 5

Find the general indefinite integral .

Solution

Question 9

Find the general indefinite integral .

Solution

Question 13 (D)

Find the general indefinite integral .

Solution

Question 34

Evaluate the integral .

Solution

Courtesy Mr. Rajesh Gunesh