sample problems for math 141 test 1 - university of kansashukle/141samplet1f10.pdf · sample...
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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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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
(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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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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