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Page 1: Sample Problems for Math 141 Test 1 - University of Kansashukle/141SampleT1f10.pdf · Sample Problems for Math 141 Test 1 MULTIPLE CHOICE (no justification necessary): 1. If fis

Sample Problems for Math 141 Test 1

MULTIPLE CHOICE (no justification necessary):

1. If f is a continuous function defined for all real numbers x and f has a maximum value f(x) = 5 anda minimum value f(x) = −7, which of the following must be true?

I. The maximum value of f(|x|) is 5 II. The maximum value of |f(x)| is 7. III the minimum value off(|x|) is 0.

(a) I only (b) I and II only (c) II only (d) II and III only (e) III only (f) I, II and III (g) I and III only

2. Let f be the function defined by f(x) =√

x+1x−2 . The domain of f is:

(a) (−∞, 2) ∪ (2,∞) (b) (−1, 2) ∪ (2,∞) (c) [−1, 2) ∪ (2,∞) (d) [1,∞)

(e) none of the above

3. If limx→2+

f(x) = π and limx→2−

f(x) = 3.14 which of the following is true?

(A) limx→2

f(x) = π and f is continuous at x = 2.

(B) limx→2

f(x) = π and f is not continuous at x = 2.

(C) limx→2

f(x) does not exist and f is continuous at x = 2.

(D) limx→2

f(x) does not exist and f is not continuous at x = 2.

(E) f is differentiable at x = 2.

4. limx→+∞

9000000x3/2 − 18000√x+ 9500

√x7 = 3100

30000x3/2 + 30√x7 − 610

√x+ 1000

is exactly

(A) 30 (B) 31 (C) 180061 (D) 316

5. If a function f satisfies the inequality 1 + cos2 x ≤ f(x) ≤ ex2−x + ex2+x for all x then lim

x→0f(x) =

(A) −∞ (B)∞ (C) 0 (D) 1 (E) 2 (F) the limit does not exist

6. If f is continuous at x = 2 with f(2) = 4 and limy→4

g(y) = 5, then limx→2g(f(x))√f(x)− 3

=

(A) −5 (B) 5 (C) 4 (D) −20 (E) None of the Above

7. The position of a particle moving along the x-axis at time t ≥ 0 (in seconds) is given by x(t) =t3 − t2 − t. In which time interval listed below is the particle slowing down?

(A) (13, 1) (B) (

13,∞) ( C) (1,∞) (D) (0, 1) (E) (1, 2)

8. The function f is continuous on the closed interval [0, 2] and has values that are given in the tablebelow. The equation f(x) = 1

2 must have at least two solutions in the interval [0, 2] if k =

x 0 1 2f(x) 1 k 2

(A) 0 (B) 12 (C) 1 (D) 2 (E) 3

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Page 2: Sample Problems for Math 141 Test 1 - University of Kansashukle/141SampleT1f10.pdf · Sample Problems for Math 141 Test 1 MULTIPLE CHOICE (no justification necessary): 1. If fis

9. The line tangent to the graph of a function f at the point (1,−2) is parallel to the line πx+ 3y = 6.

(a) f(1) = (b) f ′(1) = (c) Find an equation of the line tangent to the graph of f at (1,−2).

(d) Suppose the function f has a horizontal tangent at the point (−3,−3). Find an equation for theline tangent to the graph of f at (−3,−3).

10. The graph of the function f(x) (0 ≤ x ≤ 9) is shown below at the left. Using the grid to estimate theslope of f , sketch the graph of the derivative f ′ on the same set of axes.

y432

1

0

!1!2

1 2 9x

!3

3 4 765 8

f(x)

x

y

1

2

!1!2!3!4 1 2 3!1

3

f(x)

11. Consider the function f defined by the graph above right. Fill in the blanks. Write DNE (does notexist) if applicable.

(a) limx→1−

f(x) = (b) limx→−1

f(x) = (c) f(2) =

(g) limh→0

f(x+ h)− f(x)h

does NOT exist at x = (h) For what values of x is f

NOT continuous?

12. A particle moves along the path described by the parametric equations

x = 4 sin t and y = 1 + 2 cos t 0 ≤ t ≤ 3π/2

(a) Eliminate the parameter to get an equation in x and y. (b) Graph on your calculator and sketch thegraph. Label the points corresponding to t = 0, t = π/2, t = π, and t = 3π/2 and indicate with anarrow the path the particle traces out.

13. Let f(x) = x2 − 4x+ 6. Prove there exists at least one c ∈ (0, 3) such that f(c) = 4.

14. Let f(x) = 2x+ 2. Using an ε / δ proof, show that the limx→1

f(x) = 4.

15. Find the following limits exactly, if they exist. If the limit does not exist, write DNE. Show your workor explain your reasoning.

(a) limx→0

1− cosx√2x sinx

(b) limx→a

x2 − a2

x4 − a4(a 6= 0)

(c) limx→2|x− 2| − 2 =

(d) limx→∞

x− 23x3 −

√2x+ 21

(e) limx→3

x2 + 9x2 − 9

=

2

Page 3: Sample Problems for Math 141 Test 1 - University of Kansashukle/141SampleT1f10.pdf · Sample Problems for Math 141 Test 1 MULTIPLE CHOICE (no justification necessary): 1. If fis

(f) limx→∞

17− x2

πx2 − 17x+ 17

(g) limx→−3

x2 − 6x− 27x2 + 3x

=

(h) limx→∞

45x99 − 17x27 + 4529x27 − 35

(i) limx→0

√x+ 1− 1

x

(j) limx→3

√x2 + 7 +

√3x− 5

x+ 2

(k) limx→10−

√[[x2]]

(l) limx→∞

3− x√3x2 − 17x+ 2

(m) limx→0

sin(2θ2)θ2

(n) limx→0

sinxx+ tanx

(o) limx→0

x2 cos 10x

16. Find all asymptotes of the following functions:

(a) f(x) =x4 − x2

x(x− 1)(x+ 2)(b) f(x) =

3x2 − πxx2 − πx3

17. Let f(x) =

sinx if x ≤ 0

x2 + x if x > 0.

(a) Use the definition of continuity to prove that f is continuous at x = 0. (b) Use the definition of thederivative to determine f ′(0) if it exists.

18. Let f be differentiable everywhere and g(x) = f(ax) Prove that g′(x) = af(ax) using the limitdefinition of derivative.

19. Let f(x) =

{ax+ b x ≤ 1x4 + x+ 1 x > 1

. For what values of a and b will f be differentiable at x = 1?

20. Suppose f ′(a) = limh→0

(8 + h)2/3 − 4h

. Find a function f and a value of a that would satisfy this

equation.

21. (a) Find the slope of the line tangent to the graph of f(x) =1

2− x2at x = 2 by using the limit

definition of the derivative. (b) Find an equation of the line tangent to the graph of f at x = 2. Showyour work.

22. The distance s in feet covered by a car moving along a straight road is given by the function s(t) =2t2 + 48t. (a) Find the average velocity of the car over the time intervals [20, 21] and [20, 20]. (b)Find the instantaneous velocity of the car at time t = 20 using the definition of derivative.

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Page 4: Sample Problems for Math 141 Test 1 - University of Kansashukle/141SampleT1f10.pdf · Sample Problems for Math 141 Test 1 MULTIPLE CHOICE (no justification necessary): 1. If fis

23. The figure shows the graph of the derivative f ′ of a function f (for 0 ≤ x ≤ 9). Use the graph toanswer the following questions.

!"

#

$

%

&

!%

!$

% $ '(

!#

# " )*+ ,

-.-/0(1(A) On what interval(s) is f(x) increasing?

(B) On what interval(s) is f ′′(x) positive?

(C) Which number is larger, f(3) or f(4)?

(D) Which number is larger, f ′′(3) or f ′′(4)?

24. A function f defined and continuous on the set of all real numbers has all of the following properties.

(i) limx→−∞

f(x) = −1. (ii) Both f ′(x) and f ′′(x) are well defined and continuous at every x /∈ {3, 5},but not at x = 3 and x = 5.

(iii) f ′(x) < 0 if x < 2 or 3 < x < 5 (iv) f ′(x) > 0 if 2 < x < 3 or 5 < x

(v) f ′′(x) < 0 if x < 1 or 4 < x < 5 or 5 < x (vi)f ′′(x) > 0 if 1 < x < 3 or 3 < x < 4

(1) T F..... f is increasing on the interval (1, 3).

(2) T F..... f is concave downward on the interval (−∞, 1).

(3) T F..... f ′ is increasing on the interval (3, 4).

(4) T F..... f(2) < −1 must be true.

(5) T F..... f must have a local maximum at x = 3.

(6) T F..... The curve y = f(x) has a exactly two inflection points.

(7) T F..... y = −1 is an asymptote of the curve y = f(x).

(8) T F..... The line y = −1 must have no intersection with the curve y = f(x).

(9) T F..... The curve y = f(x) may have two different horizontal asymptotes.

(10) T F..... The line tangent to the curve y = f(x) at (4, f(4)) must be horizontal.

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