Question 1

Find the limit. $\lim _{x \rightarrow \infty} \frac{x^{5}-4}{x^{4}+5}$

Accepted Answer

A)  $\frac{7}{6}$ 
B)  -$\infty$ 
C)  $\infty$ 
D)  0 
E)  $-\frac{1}{2}$ 
A)  $\frac{7}{6}$ 
B)  -$\infty$ 
C)  $\infty$ 
D)  0 
E)  $-\frac{1}{2}$

Question 2

Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius $r=4$ .

Accepted Answer

A)  $4 \pi \sqrt{3}$ 
B)  $4 \sqrt{3}$ 
C)  $48 \sqrt{3}$ 
D)  $\sqrt{3}$ 
E)  $\sqrt{48} r^{2}$ 
A)  $4 \pi \sqrt{3}$ 
B)  $4 \sqrt{3}$ 
C)  $48 \sqrt{3}$ 
D)  $\sqrt{3}$ 
E)  $\sqrt{48} r^{2}$

Question 3

The graph below is the graph of function f on the interval $(-\infty, \infty)$ Find the absolute maximum and absolute minimum values of f (if they exist) and where they are attained.

Accepted Answer

abs. max.

Question 4

The owner of a ranch has 4000 yd of fencing with which to enclose a rectangular piece of grazing land situated along a straight portion of a river. If fencing is not required along the river, what are the dimensions of the largest area he can enclose? What is the area?

Accepted Answer

Dimensions

Question 5

Sketch the graph of the function $f(x)=\frac{x}{3} \sqrt{9-x^{2}}$ using the curve-sketching guidelines.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 6

Given $f(x)=3 x^{1 / 3}-4$ . (a) Find the intervals on which f is increasing or decreasing.
(b) Find the relative maxima and relative minima of f.

Accepted Answer

A)  (a) Increasing on $(0, \infty)$ ,
Decreasing on $(-\infty, 0)$
(b) Rel. min. $f(0)=-4$ 
B)  (a) Increasing on $(-\infty, \infty)$
(b) None 
C)  (a) Increasing on $(-\infty, 0)$ ,
Decreasing on $(0, \infty)$
(b) Rel. max. $f(0)=-4$ 
D)  (a) Decreasing on $(-\infty, \infty)$
(b) None 
A)  (a) Increasing on $(0, \infty)$ ,
Decreasing on $(-\infty, 0)$
(b) Rel. min. $f(0)=-4$ 
B)  (a) Increasing on $(-\infty, \infty)$
(b) None 
C)  (a) Increasing on $(-\infty, 0)$ ,
Decreasing on $(0, \infty)$
(b) Rel. max. $f(0)=-4$ 
D)  (a) Decreasing on $(-\infty, \infty)$
(b) None

Question 7

Given $f(x)=2 x^{3}-$$3 x^{2}-36 x+6$ . (a) Find the intervals on which f is increasing or decreasing.
(b) Find the relative maxima and relative minima of f.

Accepted Answer

A)  (a) Increasing on $(-\infty,-2)$ and $(3, \infty)$ decreasing on $(-2,3)$
(b) Rel. max. $f(-2)=50$ ,
Rel) min. $f(3)=-75$ 
B)  (a) Increasing on $(-\infty,-3)$ and $(2, \infty)$ decreasing on $(-3,2)$
(b) Rel. max. $f(-3)=33$ ,
Rel) min. $f(2)=-62$ 
C)  (a) Increasing on $(-3,2)$ , decreasing on $(-\infty,-3)$ and $(2, \infty)$
(b) Rel. max. $f(2)=62$ ,
Rel) min. $f(-3)=-33$ 
D)  (a) Increasing on $(-2,3)$ , decreasing on $(-\infty,-2)$ and $(3, \infty)$
(b) Rel. max. $f(3)=75$ ,
Rel) min. $f(-2)=-50$ 
A)  (a) Increasing on $(-\infty,-2)$ and $(3, \infty)$ decreasing on $(-2,3)$
(b) Rel. max. $f(-2)=50$ ,
Rel) min. $f(3)=-75$ 
B)  (a) Increasing on $(-\infty,-3)$ and $(2, \infty)$ decreasing on $(-3,2)$
(b) Rel. max. $f(-3)=33$ ,
Rel) min. $f(2)=-62$ 
C)  (a) Increasing on $(-3,2)$ , decreasing on $(-\infty,-3)$ and $(2, \infty)$
(b) Rel. max. $f(2)=62$ ,
Rel) min. $f(-3)=-33$ 
D)  (a) Increasing on $(-2,3)$ , decreasing on $(-\infty,-2)$ and $(3, \infty)$
(b) Rel. max. $f(3)=75$ ,
Rel) min. $f(-2)=-50$

Question 8

Find the inflection points for the function given. $f(x)=5 x+2-\sin x$ , $0<$$x<$$3 \pi$

Accepted Answer

A)  $(\pi, 5 \pi)$ , $(2 \pi, 10 \pi+2)$ 
B)  $(\pi, 5 \pi+2)$ , $(2 \pi, 10 \pi+2)$ 
C)  $(\pi, 2)$ , $(2 \pi, 10 \pi+2)$ 
D)  $(\pi, 5 \pi)$ , $(2 \pi, 10 \pi)$ 
E)  $(\pi, 5 \pi+2)$ , $(2 \pi, 10 \pi)$ 
A)  $(\pi, 5 \pi)$ , $(2 \pi, 10 \pi+2)$ 
B)  $(\pi, 5 \pi+2)$ , $(2 \pi, 10 \pi+2)$ 
C)  $(\pi, 2)$ , $(2 \pi, 10 \pi+2)$ 
D)  $(\pi, 5 \pi)$ , $(2 \pi, 10 \pi)$ 
E)  $(\pi, 5 \pi+2)$ , $(2 \pi, 10 \pi)$

Question 9

Estimate the value of $\sqrt{5}$ by using three iterations of Newton's method to solve the equation $x^{2}-5=0$ with initial estimate $x_{0}=2 \text {. }$ Round your final estimate to four decimal places.

Accepted Answer

A)  1.71 
B)  1.6535 
C)  2.2361 
D)  2.2662 
A)  1.71 
B)  1.6535 
C)  2.2361 
D)  2.2662

Question 10

Find the dimensions of the rectangle enclosed in the semicircle $y=\sqrt{144-x^{2}}$ with the largest possible area.

Accepted Answer

A)  5 in. $x$ 7 in. 
B)  $6$ in. $x$ $6$ in. 
C)  $12 \sqrt{2}$ in. $x$ $6 \sqrt{2}$ in. 
D)  $6 \sqrt{2}$ in. $x$ $6 \sqrt{2}$ in. 
A)  5 in. $x$ 7 in. 
B)  $6$ in. $x$ $6$ in. 
C)  $12 \sqrt{2}$ in. $x$ $6 \sqrt{2}$ in. 
D)  $6 \sqrt{2}$ in. $x$ $6 \sqrt{2}$ in.

Question 11

You are given the graph of a function f.
(a) Determine the intervals on which f is increasing or decreasing.
(b) Find the relative maxima and relative minima of f, if any.

Accepted Answer

(a) Increasing on @#LAT-DLM&,

Question 12

A ballast is dropped from a stationary hot-air balloon that is at an altitude of 256 ft. Find (a) an expression for the altitude of the ballast after t seconds, (b) the time when it strikes the ground, and (c) its velocity when it strikes the ground. (Disregard air resistance and take $ g=32 \mathrm{ft} / \sec ^{2} $ .)

Accepted Answer

The answer of A ballast is dropped from a stationary...

Question 13

Find the value of the limit. $\lim _{x \rightarrow \infty} \frac{x^{3}}{3^{x}}$

Accepted Answer

The answer of Find the value of the limit. \(\lim...

Question 14

Evaluate the limit using l'Hôpital's Rule. $\lim _{x \rightarrow 5} \frac{x^{3}-125}{x-5}$

Accepted Answer

A)  25 
B)  0 
C)  75 
D)  15 
A)  25 
B)  0 
C)  75 
D)  15

Question 15

Evaluate $f(x)=\sin \left(x^{2}\right)$ , and tell whether its antiderivative F is increasing or decreasing at the point $x=-4$ radians.

Accepted Answer

A)  - 0.288, increasing 
B)  0.757, decreasing 
C)  $-0.757$ , decreasing 
D)  0.757, increasing 
E)  0.277, decreasing 
A)  - 0.288, increasing 
B)  0.757, decreasing 
C)  $-0.757$ , decreasing 
D)  0.757, increasing 
E)  0.277, decreasing

Question 16

A poster of height 33 in. is mounted on a wall so that its lower edge is 15 in. above the eye level of an observer. How far from the wall should the observer stand so that the viewing angle $\theta$ subtended at his eye by the poster is as large as possible (see figure - not drawn to scale)?   Round your answer to the nearest integer.

Accepted Answer

The answer of A poster of height 33 in. is...

Question 17

Find
f.
 $f^{\prime \prime}(x)=x^{3}+\sinh x, f(0)=1, f(2)=\frac{13}{5}$

Accepted Answer

The answer of Find
f.
 $f^{\prime \prime}(x)=x^{3}+\sinh x, f(0)=1, f(2)=\frac{13}{5}$...

Question 18

Find the limit. $\lim _{u \rightarrow 2^{+}} \frac{-5 u^{2}}{u-2}$

Accepted Answer

A)  $\infty$ 
B)  -$\infty$ 
C)  $-\frac{5}{2}$ 
D)  $\frac{5}{2}$ 
A)  $\infty$ 
B)  -$\infty$ 
C)  $-\frac{5}{2}$ 
D)  $\frac{5}{2}$

Question 19

Evaluate the limit. $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{5}+x^{4}}$

Accepted Answer

A)  0 
B)  $\infty$ 
C)  -5 
D)  1 
E)  $\pi$ 
A)  0 
B)  $\infty$ 
C)  -5 
D)  1 
E)  $\pi$

Question 20

Find the maximum or minimum point(s) of the function. $F(x)=\left(1-x^{2}\right)^{2}+6 x^{2}$

Accepted Answer

A)  $(16,1)$ 
B)  $(-8,2)$ 
C)  $(0,1)$ 
D)  $(8,0)$ 
E)  $(-8.6,6)$ 
A)  $(16,1)$ 
B)  $(-8,2)$ 
C)  $(0,1)$ 
D)  $(8,0)$ 
E)  $(-8.6,6)$

Find the limit. $\lim _{x \rightarrow \infty} \frac{x^{5}-4}{x^{4}+5}$

Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius $r=4$ .

The owner of a ranch has 4000 yd of fencing with which to enclose a rectangular piece of grazing land situated along a straight portion of a river. If fencing is not required along the river, what are the dimensions of the largest area he can enclose? What is the area?

Sketch the graph of the function $f(x)=\frac{x}{3} \sqrt{9-x^{2}}$ using the curve-sketching guidelines.

Given $f(x)=3 x^{1 / 3}-4$ . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

Given $f(x)=2 x^{3}-$ $3 x^{2}-36 x+6$ . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

Find the inflection points for the function given. $f(x)=5 x+2-\sin x$ , $0<$ $x<$ $3 \pi$

Estimate the value of $\sqrt{5}$ by using three iterations of Newton's method to solve the equation $x^{2}-5=0$ with initial estimate $x_{0}=2 \text {. }$ Round your final estimate to four decimal places.

Find the dimensions of the rectangle enclosed in the semicircle $y=\sqrt{144-x^{2}}$ with the largest possible area. $Find the dimensions of the rectangle enclosed in the semicircle y=\sqrt{144-x^{2}} with the largest possible area.$

You are given the graph of a function f. (a) Determine the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f, if any.

Find the value of the limit. $\lim _{x \rightarrow \infty} \frac{x^{3}}{3^{x}}$

Evaluate the limit using l'Hôpital's Rule. $\lim _{x \rightarrow 5} \frac{x^{3}-125}{x-5}$

Evaluate $f(x)=\sin \left(x^{2}\right)$ , and tell whether its antiderivative F is increasing or decreasing at the point $x=-4$ radians.

Find f. $f^{\prime \prime}(x)=x^{3}+\sinh x, f(0)=1, f(2)=\frac{13}{5}$

Find the limit. $\lim _{u \rightarrow 2^{+}} \frac{-5 u^{2}}{u-2}$

Evaluate the limit. $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{5}+x^{4}}$

Find the maximum or minimum point(s) of the function. $F(x)=\left(1-x^{2}\right)^{2}+6 x^{2}$

Functions and Limits

Derivatives

Applications of Differentiation

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 4: Integrals

Find the limit. lim⁡x→∞x5−4x4+5\lim _{x \rightarrow \infty} \frac{x^{5}-4}{x^{4}+5}limx→∞​x4+5x5−4​

Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius r=4r=4r=4 .

The graph below is the graph of function f on the interval (−∞,∞)(-\infty, \infty)(−∞,∞) Find the absolute maximum and absolute minimum values of f (if they exist) and where they are attained.

The owner of a ranch has 4000 yd of fencing with which to enclose a rectangular piece of grazing land situated along a straight portion of a river. If fencing is not required along the river, what are the dimensions of the largest area he can enclose? What is the area?

Sketch the graph of the function f(x)=x39−x2f(x)=\frac{x}{3} \sqrt{9-x^{2}}f(x)=3x​9−x2​ using the curve-sketching guidelines.

Given f(x)=3x1/3−4f(x)=3 x^{1 / 3}-4f(x)=3x1/3−4 . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

Given f(x)=2x3−f(x)=2 x^{3}-f(x)=2x3−3x2−36x+63 x^{2}-36 x+63x2−36x+6 . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

Find the inflection points for the function given. f(x)=5x+2−sin⁡xf(x)=5 x+2-\sin xf(x)=5x+2−sinx , 0<0<0<x<x<x<3π3 \pi3π

Estimate the value of 5\sqrt{5}5​ by using three iterations of Newton's method to solve the equation x2−5=0x^{2}-5=0x2−5=0 with initial estimate x0=2. x_{0}=2 \text {. }x0​=2. Round your final estimate to four decimal places.

Find the dimensions of the rectangle enclosed in the semicircle y=144−x2y=\sqrt{144-x^{2}}y=144−x2​ with the largest possible area.

You are given the graph of a function f. (a) Determine the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f, if any.

Find the value of the limit. lim⁡x→∞x33x\lim _{x \rightarrow \infty} \frac{x^{3}}{3^{x}}limx→∞​3xx3​

Evaluate the limit using l'Hôpital's Rule. lim⁡x→5x3−125x−5\lim _{x \rightarrow 5} \frac{x^{3}-125}{x-5}limx→5​x−5x3−125​

Evaluate f(x)=sin⁡(x2)f(x)=\sin \left(x^{2}\right)f(x)=sin(x2) , and tell whether its antiderivative F is increasing or decreasing at the point x=−4x=-4x=−4 radians.

Find f. f′′(x)=x3+sinh⁡x,f(0)=1,f(2)=135f^{\prime \prime}(x)=x^{3}+\sinh x, f(0)=1, f(2)=\frac{13}{5}f′′(x)=x3+sinhx,f(0)=1,f(2)=513​

Find the limit. lim⁡u→2+−5u2u−2\lim _{u \rightarrow 2^{+}} \frac{-5 u^{2}}{u-2}limu→2+​u−2−5u2​

Evaluate the limit. lim⁡x→01−cos⁡xx5+x4\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{5}+x^{4}}limx→0​x5+x41−cosx​

Find the maximum or minimum point(s) of the function. F(x)=(1−x2)2+6x2F(x)=\left(1-x^{2}\right)^{2}+6 x^{2}F(x)=(1−x2)2+6x2

Functions and Limits

Derivatives

Applications of Differentiation

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Find the limit. $\lim _{x \rightarrow \infty} \frac{x^{5}-4}{x^{4}+5}$

Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius $r=4$ .

Sketch the graph of the function $f(x)=\frac{x}{3} \sqrt{9-x^{2}}$ using the curve-sketching guidelines.

Given $f(x)=3 x^{1 / 3}-4$ . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

Given $f(x)=2 x^{3}-$ $3 x^{2}-36 x+6$ . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

Find the inflection points for the function given. $f(x)=5 x+2-\sin x$ , $0<$ $x<$ $3 \pi$

Estimate the value of $\sqrt{5}$ by using three iterations of Newton's method to solve the equation $x^{2}-5=0$ with initial estimate $x_{0}=2 \text {. }$ Round your final estimate to four decimal places.

Find the dimensions of the rectangle enclosed in the semicircle $y=\sqrt{144-x^{2}}$ with the largest possible area. $Find the dimensions of the rectangle enclosed in the semicircle y=\sqrt{144-x^{2}} with the largest possible area.$

Find the value of the limit. $\lim _{x \rightarrow \infty} \frac{x^{3}}{3^{x}}$

Evaluate the limit using l'Hôpital's Rule. $\lim _{x \rightarrow 5} \frac{x^{3}-125}{x-5}$

Evaluate $f(x)=\sin \left(x^{2}\right)$ , and tell whether its antiderivative F is increasing or decreasing at the point $x=-4$ radians.

Find f. $f^{\prime \prime}(x)=x^{3}+\sinh x, f(0)=1, f(2)=\frac{13}{5}$

Find the limit. $\lim _{u \rightarrow 2^{+}} \frac{-5 u^{2}}{u-2}$

Evaluate the limit. $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{5}+x^{4}}$

Find the maximum or minimum point(s) of the function. $F(x)=\left(1-x^{2}\right)^{2}+6 x^{2}$