2 Sample Exam Questions

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CHAPTER 2 SAMPLE EXAM QUESTIONS. 8. Give examples of functions f (x) and g (x) with lim x→∞ f (x) = ∞, lim x→∞ g (x) = ∞ and. (a) lim x→∞ f (x) g (x).
2 Sample Exam Questions Problems marked with an asterisk (*) are particularly challenging and should be given careful consideration. 1. Consider the following graph of f .

4 3 2 1 4 1

2

3

5

t

(a) What is lim+ f (t)? lim− f (t)? lim− f (t)? lim f (t)? t→0

t→0

t→2

t→−∞

(b) For what values of x does lim f (t) exist? t→x

(c) Does f have any vertical asymptotes? If so, where? (d) Does f have any horizontal asymptotes? If so, where? (e) For what values of x is f discontinuous? 2. Find values for a and b that will make f continuous everywhere, if

⎧ ⎪ ⎨ 3x + 1 f (x) = ax + b ⎪ ⎩ 2 x

if x < 2 if 2 ≤ x < 5 if 5 ≤ x  −1 3. Find the vertical and horizontal asymptotes for f (x) = a−1 + x−1 , where a is a positive number. 4. Consider the function f (x) =

x+4 . x2 + 3x − 4

(a) What is the domain of f ? (b) Compute lim f (x), if this limit exists. x→−4

(c) Is f continuous at x = −4? Explain your answer by either proving that f is continuous at x = −4 or telling how to modify f to make it continuous. 147

CHAPTER 2 LIMITS AND DERIVATIVES

5. Let f be a continuous function such that f (−1) = −1 and f (1) = 1. Classify the following statements

as A. Always true B. Never true C. Sometimes true Justify your answers. (a) f (0) = 0 (b) For some x with −1 ≤ x ≤ 1, f (x) = 0 (c) For all x with −1 ≤ x ≤ 1, −1 ≤ f (x) ≤ 1 (d) Given any y in [−1, 1], then y = f (x) for some x in [−1, 1]. (e) If x < −1 or x > 1, then f (x) < −1 or f (x) > 1. (f) f (x) = −1 for x < 0 and f (x) = 1 for x > 0. 6. Let f be the function whose graph is given below. y 2

f 1

0

1

2

(a) Sketch a plausible graph of f  .

3

x

(b) Sketch a plausible graph of a function F such that F  = f and F (0) = 1.

2

3

1

2 1

_1 _2

7. Suppose that the line tangent to the graph of y = f (x) at x = 3 passes through the points (−2, 3) and

(4, −1).

(a) Find f  (3). (b) Find f (3). (c) Find an equation of the line tangent to f at x = 3. 148

CHAPTER 2 SAMPLE EXAM QUESTIONS

8. Give examples of functions f (x) and g (x) with lim f (x) = ∞, lim g (x) = ∞ and x→∞

x→∞

f (x) (a) lim =∞ x→∞ g (x) f (x) =6 x→∞ g (x)

(b) lim

(c) lim

x→∞

f (x) =0 g (x) f (x) = −1? Either give an example or explain why it is not possible. x→∞ g (x)

(d) Is it possible to have lim

9. Each of the following limits represent the derivative of a function f at some point a. State a formula for

f and the value of the point a. (3 + h)2 − 9 h→0 h

2x − 2 x→1 x − 1

(b) lim

(a) lim

(x + 1)3/2 − 8 x→3 x−3

sin (π (2 + h)) − 0 h→0 h

(c) lim

(d) lim

10. Let

⎧ √ ⎪ ⎨ 3−x f (x) = x2 ⎪ ⎩ 27/x

if x ≤ 1 if 1 < x < 3 if x ≥ 3

(a) Evaluate each limit, if it exists. (i) lim− f (x)

(ii) lim+ f (x)

(iii) lim f (x)

(iv) lim− f (x)

(v) lim+ f (x)

(vi) lim f (x)

(vii) lim f (x)

(viii) lim f (x)

x→1 x→3

x→1

x→1 x→3

x→9

x→3

x→−6

(b) Where is f discontinuous? 11. The graph of f (x) is given below.

For which value(s) of x is f (x) not differentiable? Justify your

answer(s).

149

CHAPTER 2 LIMITS AND DERIVATIVES

12. A bicycle starts from rest and its distance travelled is recorded in the following table at one-second

intervals. t (s) d (ft)

0 0

1 10

2 24

3 42

4 63

5 84.5

6 107

(a) Compute the average speed of the bicycle from t = 1 to t = 2 and from t = 2 to t = 3. (b) Estimate the speed after 2 seconds. (c) Estimate the speed after 5 seconds. (d) Estimate the speed after 6 seconds. (e) Can we determine if the cyclist’s speed is constantly increasing? Explain. 13. Referring to the graphs given below, find each limit, if it exists. If a limit does not exist, explain why not.

f (x) x→0 g (x)

(a) lim

g (x) x→−1 f (x) g (x) (g) lim x→1 f (x)

(b) lim [g (x) · f (x)] x→1

(e) lim [g (x) + f (x)] (f) lim− [x + f (x)] x→−1

(d) lim [x · g (x)]

(c) lim

x→1

x→2

14. The following is a graph of f  , the derivative of some function f . y 2 1

_2

_1

f» 1

2

x

_1 _2

(a) Where is f increasing? (b) Where does f have a local minimum? Where does f have a local maximum? (c) Where is f concave up? (d) Assuming that f (0) = −1, sketch a possible graph of f . 150

CHAPTER 2 SAMPLE EXAM SOLUTIONS

2 Sample Exam Solutions 1. (a) lim f (t) = ∞, lim f (t) = 1, lim f (t) = 3, lim f (t) = 1 t→0+

t→0−

t→−∞

t→2−

(b) lim f (t) exists for all x except x = 0 and x = 2. t→x

(c) There is a vertical asymptote at x = 0. (d) There is a horizontal asymptote at y = 1. (e) f is discontinuous at x = 0, 2, and 4. 2. Solve 3 (2) + 1 = 2a + b and 52 = 5a + b to get a = 6, b = −5. 3. Taking lim f (x) gives a horizontal asymptote at y = a. Algebraic simplification gives a vertical x→∞

asymptote at x = −a. The function is undefined at x = 0, but there is no asymptote there because lim f (x) = 0.

x→0

4. f (x) =

x+4 (x + 4) (x − 1)

(a) The domain is all values of x except x = 1 and x = −4. (b) Algebraic simplification gives a limit of − 15 . (c) f is not continuous at x = −4, for it is not defined there. It can be modified by defining f (−4) to be − 15 . 5. (a) C. True for f (x) = x, untrue for f (x) = x2 + x − 1

(b) A. True by the Intermediate Value Theorem (c) C. True for f (x) = x, untrue for f (x) = x2 + x − 1 (d) A. True by the Intermediate Value Theorem (e) C. True for f (x) = x, untrue for f (x) = x2 + x − 1 (f) B. lim f (x) does not exist, contradicting the continuity of f . x→0

6. (a) Answers will vary. Look for:

(i) zeros at 1 and 2

(iii) f  negative for x ∈ (1, 2)

(ii) f  positive for x ∈ (0, 1) and (2, 4)

(iv) f  flattens out for x > 2.5

(b) Answers will vary. Look for (i) F (0) = 1

(iv) F is closest to being flat at x = 2

(ii) F is always increasing

(v) F is concave up for x ∈ (0, 1) and x ∈ (2, 4)

(iii) F is never perfectly flat

(vi) F is concave down for x ∈ (1, 2) 151

CHAPTER 2 LIMITS AND DERIVATIVES

7. (a)

3 − (−1) 2 =− −2 − 4 3

(b) The equation of the tangent line is y − 3 = − 23 (x + 2), so f (3) = − 23 (3 + 2) + 3 = − 13 . (c) The equation of the tangent line is y − 3 = − 23 (x + 2). 8. Answers will vary; the following are samples only.

(a) f (x) = x2 , g (x) = x (b) f (x) = 6x, g (x) = x (c) f (x) = x, g (x) = x2 f (x) = −1, either f or g would have to be negative for large x. This g (x) contradicts the assumption that lim f (x) = lim g (x) = ∞.

(d) This is not possible. For lim

x→∞

x→∞

x→∞

9. Answers will vary.

(a) f (x) = x2 , a = 3

(b) f (x) = 2x , a = 1

(c) f (x) = (x + 1)3/2 , a = 3 √ 10. (a) (i) 2 (ii) 1 (iii) Does not exist

(d) f (x) = sin (πx), a = 2 (iv) 9

(v) 9

(vi) 9

(vii) 3

(viii) 3

(b) f is discontinuous at x = 1. 11. f isn’t differentiable at x = 1, because it is not continuous there; at x = −2, because it has a vertical

tangent there; and at x = 4, because it has a cusp there. 12. (a) Answers will vary.

One good answer would be to compute the average speed between 1 and 2 (14 ft/s) and the average speed between 2 and 3 (18 ft/s) and average them to get 16 ft/s. This is also the answer obtained by computing the average speed between 1 and 3.

(b) Answers will vary. Using reasoning similar to the previous part, we get an estimate of 22 ft/s, but it could be argued that a number closer to 22.5 would be more accurate. (c) Answers will vary. The average speed between t = 5 and t = 6 is 22.5 ft/s. (d) Since we are given information only about the cyclist’s position at one-second intervals, we cannot determine if the speed is constantly increasing. 13. (a)

1 2

(b) 0

(d) −4

(c) Does not exist, because lim f (x) = 0 while lim g (x) = 1. x→−1

(e) 1

(f) 2

14. (a) f is increasing on (−1, 1).

(c) f is concave up where

(g) 0

(d)

(b) Local minimum at x = −1; local maximum at x = 1 f  (x)

x→−1

y _2

_1

0

f 1

_1 Inflection point

is

increasing, that is, on (−2, 0).

_2

152

2 x