Description

Questions

1. Define the function h(x)=∫1xcos⁡(t2) dth(x)=∫1x​​cos(t2)dt. Find h′(4)h′(4).

(A) 14(cos⁡(4)−cos⁡(1))41​(cos(4)−cos(1))
(B) 14(sin⁡(4)−sin⁡(1))41​(sin(4)−sin(1))
(C) $\cos (4)$
(D) cos⁡(4)44cos(4)​

2. Suppose a function $f$ satisfies

x2<f(x)<12x(x+1)x2<f(x)<21​x(x+1), for $0<x<1$
$f(x)=2x+1$, for $x\ge 1$.

Which of the following statements must be true?

(A) lim⁡x→1−f(x)limx→1−​f(x) exists
(B) $f(0)=0$
(C) lim⁡x→12f(x)limx→21​​f(x) does not exist
(D) $f$ is continuous at $x=1$

3. The graph of $y=f(x)$ is given below. If Newton’s method is applied to the function $f$, starting with an initial guess of x1=1x1​=1, what happens to the iterates x2,x3,x4,…x2​,x3​,x4​,…?

[Graph image — insert here]

(A) x2=1.5x2​=1.5, x3=3x3​=3, and x4=6.75x4​=6.75.
(B) x2=6x2​=6 and x3x3​ cannot be calculated because f′(x2)=0f′(x2​)=0.
(C) The iterates x2,x3,x4,…x2​,x3​,x4​,… oscillate and do not approach any root.
(D) The iterates x2,x3,x4,…x2​,x3​,x4​,… will approach the root $x=5$.

4. The linearization of a function $f$ at $a=2$ is given by L(x)=12xL(x)=21​x. One of the choices below is the graph of $f$. Choose the correct graph of $f$.

[Graph images (A)–(D) — insert here]

5. Express the following integral as a limit of Riemann sums: ∫25(x2+1) dx∫25​(x2+1)dx

(A) lim⁡n→∞∑i=1n((2+in)2+1)1nlimn→∞​∑i=1n​((2+ni​)2+1)n1​
(B) lim⁡n→∞∑i=1n((2+3in)2+1)3nlimn→∞​∑i=1n​((2+n3i​)2+1)n3​
(C) lim⁡n→∞∑i=1n((in)2+1)3nlimn→∞​∑i=1n​((ni​)2+1)n3​
(D) lim⁡n→∞∑i=1n(3+3in)23nlimn→∞​∑i=1n​(3+n3i​)2n3​

6. After an appropriate $u$-substitution, which of the following integrals is equivalent to ∫1e(ln⁡(x))2x dx∫1e​x(ln(x))2​dx?

(A) ∫01u2eu du∫01​euu2​du
(B) ∫1eu2 du∫1e​u2du
(C) ∫01u2 du∫01​u2du
(D) ∫1euln⁡(u) du∫1e​uln(u)du

7. Two people, $A$ and $B$, are standing 3 miles apart on a straight road. At the same time that person $A$ walks north at a rate of 3 miles per hour (mph), person $B$ walks south at a rate of 1 mph. At what rate is the distance between the two people changing 1 hour later?

(A) 1010​ mph
(B) 165516​ mph
(C) 5225​ mph
(D) 163316​ mph

[Diagram — insert here]

8. If the function $f$ is differentiable on $[a,b]$, then which of the following statements must be true?

I. $f$ has a maximum value on the open interval $(a,b)$.
II. f′(c)=f(b)−f(a)b−af′(c)=b−af(b)−f(a)​ for some $c$ such that $a<c<b$.
III. f′(c)=0f′(c)=0 for some $c$ such that $a<c<b$.

(A) Only I
(B) Only II
(C) Only I and III
(D) Only II and III

9. Consider the graph of $y=f(x)$ below, and suppose that ∫28f(x) dx=11∫28​f(x)dx=11.

[Graph image — insert here]

The value of ∫48(2f(x)+3) dx∫48​(2f(x)+3)dx is

(A) 26
(B) 22
(C) 17
(D) 7

10. Suppose

f′(a)=lim⁡h→08+6h+h23−2hf′(a)=limh→0​h38+6h+h2​−2​.

From the choices below, choose the correct function $f$ and the corresponding value of $a$.

(A) f(x)=x2+13f(x)=3x2+1​, $a=0$
(B) f(x)=x23+2f(x)=3x2​+2, $a=0$
(C) f(x)=x2−13f(x)=3x2−1​, $a=3$
(D) f(x)=1−x23f(x)=1−3x2​, $a=3$

11. Consider the equation y2+x2=3xy−4y2+x2=3xy−4. Find dydxdxdy​ evaluated at the point $(2,4)$.

(A) $-2$
(B) −45−54​
(C) 1441​
(D) 4

12. Let $f$ be a function such that f′′f′′ is continuous on $(-\infty ,+\infty )$, f′(4)=0f′(4)=0 and f′′(4)=−1f′′(4)=−1. Consider g(x)=f(x)+(x−4)2g(x)=f(x)+(x−4)2. Which of the following can be determined about $g$ at $x=4$?

(A) $g$ has a local minimum at $x=4$.
(B) $g$ has a local maximum at $x=4$.
(C) $g$ has an inflection point at $x=4$.
(D) Nothing can be determined about $g$.

13. Consider the graph of $y=f(x)$ below. Which of the following statements is FALSE?

[Graph image — insert here]

(A) The line $x=4$ is a vertical asymptote.
(B) lim⁡x→21f(x)=0limx→2​f(x)1​=0
(C) lim⁡x→4+f(x)=1limx→4+​f(x)=1
(D) lim⁡x→6f(x)=−∞limx→6​f(x)=−∞

14. Consider the graph of $y=f(x)$ below. Find the largest value of $\delta >0$ such that if $0<|x-4|<\delta$, then $|f(x)-2|<1$.

[Graph image — insert here]

(A) $\delta =1$
(B) $\delta =2$
(C) $\delta =3$
(D) no such $\delta$ exists because $f$ is not continuous

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1225ctf_f16.pdf — Form A

Instructions

Fill in A, B or C in the Test Version section.

Enter your NAME, ID Number, CRN (under Class ID) and write A, B, or C (under Test ID) on the op-scan sheet.

Darken the appropriate circles below your ID number and Class ID (CRN). Use a number 2 pencil. Machine grading may ignore faintly marked circles.

Mark your answers to the test questions in rows 1–14 of the op-scan sheet. Your score on this test will be the number of correct answers.

You have one hour to complete this portion of the exam. Turn in the op-scan sheet with your answers, this exam and all scrap paper at the end of this part of the final exam.

Exam Policies: You may not use a book, notes, formula sheet, or a calculator or computer.

Name (printed): __________________________

Student ID #: __________________________

Honor Pledge: I have neither given nor received unauthorized assistance on this exam.

Signature: __________________________

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Questions

1. Consider the vertical and horizontal asymptotes of the following function:

[Function image missing in source — insert here]

Which one of the following statements is TRUE?

(A) $f(x)$ has exactly one vertical asymptote and one horizontal asymptote.
(B) $f(x)$ has exactly one vertical asymptote and two horizontal asymptotes.
(C) $f(x)$ has exactly two vertical asymptotes and one horizontal asymptote.
(D) $f(x)$ has exactly two vertical asymptotes and two horizontal asymptotes.

2. Let $f(x)=\cos x$. Find f(8)(x)−f(5)(x)f(8)(x)−f(5)(x).

(A) $2\sin x$
(B) $\cos x+\sin x$
(C) $\sin x-\cos x$
(D) 0

3. Find the linearization $L(x)$ centered at $x=e$ for f(x)=ln⁡xf(x)=lnx​.

(A) L(x)=1+x2eL(x)=1+2ex​
(B) L(x)=x2e−14L(x)=2ex​−41​
(C) and (D) missing in source — insert here

4. Which of the following functions

f(x)=x2/3f(x)=x2/3 on $[-1,8]$, g(x)=x4/5g(x)=x4/5 on $[0,1]$

satisfy the hypotheses of the Mean Value Theorem on their respective intervals?

(A) Only $f(x)$.
(B) Only $g(x)$.
(C) Both $f(x)$ and $g(x)$.
(D) Neither $f(x)$ nor $g(x)$.

5. The graph of a function $f(x)$ on $[0,4]$ is given below. Find the largest value of $\delta >0$ such that for all $x$ with $0<|x-3|<\delta$ implies $|f(x)-1.5|<1$

[Graph image — insert here]

(A) $\delta =0.5$
(B) $\delta =1$
(C) $\delta =1.5$
(D) $\delta =3$

6. Evaluate the indefinite integral: ∫3×5−2x7x6 dx∫x63x5−2x7​dx

(A) 12×6−14x817x7+C71​x721​x6−41​x8​+C
(B) (12×6−14×8)ln⁡(x6)+C(21​x6−41​x8)ln(x6)+C
(C) 15x−5(12×6−14×8)+C51​x−5(21​x6−41​x8)+C
(D) 3ln⁡(x)−x2+C3ln(x)−x2+C

7. You want to minimize the cost of building a rectangular box with capacity of 100 cubic meters. The box must have a square top and bottom, with edge length $x$, as in the picture below. The material for the sides costs $2 per square meter, and for the top and bottom costs $3 per square meter. Which function should be minimized?

[Box diagram — insert here]

(A) f(x)=x(200−6×2)f(x)=x(200−6x2)
(B) f(x)=x(200−6×2)−100f(x)=x(200−6x2)−100
(C) f(x)=6×2+800xf(x)=6x2+x800​
(D) f(x)=x(100−6×2)−200f(x)=x(100−6x2)−200

8. The graphs of $f$ and $g$ are given below. Determine which limit does NOT exist.

[Graphs of f and g — insert here]

(A) lim⁡x→1f(x)g(x)limx→1​f(x)g(x)
(B) lim⁡x→2g(x)f(x)limx→2​f(x)g(x)​
(C) lim⁡x→3f(x)g(x)limx→3​g(x)f(x)​
(D) All these limits exist

9. The tangent line to the ellipse 5×2−4xy+5y2=15x2−4xy+5y2=1 at the point (0,15)(0,5​1​) has slope:

(A) 4555+255​+245​​
(B) 2552​
(C) 1
(D) The tangent line is vertical

10. Let $f$ and $g$ be continuous functions, and suppose that ∫02f(x) dx=5∫02​f(x)dx=5 and ∫02g(x) dx=7∫02​g(x)dx=7. Find ∫022f(x)−3g(x)+10 dx∫02​2f(x)−3g(x)+10dx.

(A) 9
(B) 51
(C) 8
(D) -1

11. What is TRUE about the function F(x)=∫5sin⁡(x)et2 dtF(x)=∫5sin(x)​et2dt?

(A) $F(x)$ is an increasing function for all $x$ values.
(B) F′(x)=esin⁡2(x)−5F′(x)=esin2(x)−5.
(C) $F(x)$ is decreasing on the interval $(\pi /2,\pi )$.
(D) $F(x)$ is a decreasing function for all $x$ values.

12. Consider the following function

[Function image missing in source — insert here]

Which one of the following statements is TRUE?

(A) $f$ has a critical number at $x=1$.
(B) The absolute maximum occurs at $x=1$.
(C) $f$ has a local maximum at $x=1$.
(D) $f$ is continuous at $x=1$.

13. Use Newton’s method with initial approximation x1=2x1​=2 to find x2x2​, the second approximation to a root of f(x)=x2−3f(x)=x2−3.

(A) 3223​
(B) 9449​
(C) 3443​
(D) 7447​

14. When you use the upper Riemann sum with $n=4$ sub-intervals to estimate ∫−44∣x∣ dx∫−44​∣x∣dx, you get:

(A) 0
(B) 16
(C) 24
(D) 48