Question 1

Find the derivative of the function. $f(x)=2 \sqrt{x}+4 e^{x}$

Accepted Answer

The answer of Find the derivative of the function. \(f(x)=2...

Question 2

Find the derivative of $f(x)$ . $f(x)=x^{8} \cosh x$

Accepted Answer

The answer of Find the derivative of $f(x)$ . \(f(x)=x^{8}...

Question 3

Find the second derivative of the function. $f(x)=5 e^{x} \cos x$

Accepted Answer

A)  $10 e^{x} \sin x$ 
B)  $10 e^{x} \cos x$ 
C)  $-10 e^{x} \cos x$ 
D)  $-10 e^{x} \sin x$ 
A)  $10 e^{x} \sin x$ 
B)  $10 e^{x} \cos x$ 
C)  $-10 e^{x} \cos x$ 
D)  $-10 e^{x} \sin x$

Question 4

The function $f(t)=\frac{t}{t-8}$ satisfies the hypotheses of the Mean Value Theorem on the interval $[-2,0]$ . Find all values of c that satisfy the conclusion of the theorem.

Accepted Answer

The answer of The function $f(t)=\frac{t}{t-8}$ satisfies the hypotheses of...

Question 5

Find an equation of the tangent line to the curve $90\left(x^{2}+y^{2}\right)^{2}=1734\left(x^{2}-y^{2}\right)$ at the point (4,1).

Accepted Answer

A)  $y=1.11 x+5.43$ 
B)  $y=-1.11 x+17$ 
C)  $y=-1.11 x+3.43$ 
D)  $y=-1.11 x+5.43$ 
E)  $\text { None of these }$ 
A)  $y=1.11 x+5.43$ 
B)  $y=-1.11 x+17$ 
C)  $y=-1.11 x+3.43$ 
D)  $y=-1.11 x+5.43$ 
E)  $\text { None of these }$

Question 6

Find the derivative of the function. $f(u)=e^{u} \cot u$

Accepted Answer

A)  $e^{u}\left(\cot u+\csc ^{2} u\right)$ 
B)  $-e^{u} \csc ^{2} u$ 
C)  $e^{u} \csc ^{2} u$ 
D)  $e^{u}\left(\cot u-\csc ^{2} u\right)$ 
A)  $e^{u}\left(\cot u+\csc ^{2} u\right)$ 
B)  $-e^{u} \csc ^{2} u$ 
C)  $e^{u} \csc ^{2} u$ 
D)  $e^{u}\left(\cot u-\csc ^{2} u\right)$

Question 7

Determine the values of x for which the given linear approximation is accurate to within 0.07 at a = 0. $\tan x \approx X$

Accepted Answer

A)  $-0.19<$$x<0.28$ 
B)  $-0.57<$$x<0.57$ 
C)  $0.06<$$x<0.68$ 
D)  $-1.04<$$x<1.55$ 
E)  $-0.71<$$x<0.48$ 
A)  $-0.19<$$x<0.28$ 
B)  $-0.57<$$x<0.57$ 
C)  $0.06<$$x<0.68$ 
D)  $-1.04<$$x<1.55$ 
E)  $-0.71<$$x<0.48$

Question 8

s(t) is the position of a body moving along a coordinate line, where $t \geq 0$ , and s(t) is measured in feet and t in seconds. $s(t)=1+6 t-t^{2}$ 
a. Determine the time(s) and the position(s) when the body is stationary.
b. When is the body moving in the positive direction? In the negative direction?
c. Sketch a schematic showing the position of the body at any time t.

Accepted Answer

a. @#LAT-DLM&
b. Pos

Question 9

Find an equation of the tangent line to the curve.
 $y=\frac{\sqrt{x}}{x+6} \text { at }(4,0.2)$

Accepted Answer

The answer of Find an equation of the tangent line...

Question 10

Find dy/dx by implicit differentiation. $5 x^{2}+9 y^{2}=5$

Accepted Answer

A)  $\frac{5-10 x}{18 y}$ 
B)  $-\frac{5 x}{9 y}$ 
C)  $-\frac{5 x}{9}$ 
D)  $\frac{5-10 x}{18}$ 
A)  $\frac{5-10 x}{18 y}$ 
B)  $-\frac{5 x}{9 y}$ 
C)  $-\frac{5 x}{9}$ 
D)  $\frac{5-10 x}{18}$

Question 11

The altitude of a triangle is increasing at a rate of $1 \mathrm{~cm} / \min$ while the area of the triangle is increasing at a rate of $2 \mathrm{~cm}^{2} / \min$ . At what rate is the base of the triangle changing when the altitude is 10 cm and the area is $200 \mathrm{~cm}^{2}$ .

Accepted Answer

The answer of The altitude of a triangle is increasing...

Question 12

Find an equation of the tangent line to the curve $x e^{y}+x+2 y=2$ at (1, 0).

Accepted Answer

The answer of Find an equation of the tangent line...

Question 13

Calculate $y^{\prime}$ . $x y^{4}+x^{2} y=5 x+3 y$

Accepted Answer

The answer of Calculate $y^{\prime}$ . \(x y^{4}+x^{2} y=5 x+3...

Question 14

Differentiate the function. $B(y)=c y^{-4}$

Accepted Answer

A)  $B^{\prime}(y)=\frac{5 c}{y^{4}}$ 
B)  $B^{\prime}(y)=-\frac{5 c}{y^{4}}$ 
C)  $B^{\prime}(y)=-\frac{4 c^{5}}{y}$ 
D)  $B^{\prime}(y)=-\frac{4 c}{y^{5}}$ 
E)  $B^{\prime}(y)=-\frac{c}{4 y^{5}}$ 
A)  $B^{\prime}(y)=\frac{5 c}{y^{4}}$ 
B)  $B^{\prime}(y)=-\frac{5 c}{y^{4}}$ 
C)  $B^{\prime}(y)=-\frac{4 c^{5}}{y}$ 
D)  $B^{\prime}(y)=-\frac{4 c}{y^{5}}$ 
E)  $B^{\prime}(y)=-\frac{c}{4 y^{5}}$

Question 15

Find the first and the second derivatives of the function. $y=\frac{x}{5-x}$

Accepted Answer

The answer of Find the first and the second derivatives...

Question 16

Find the derivative of the function. $f(x)=-x^{2}+7 x+5$

Accepted Answer

The answer of Find the derivative of the function. \(f(x)=-x^{2}+7...

Question 17

Two sides of a triangle are 2 m and 3 m in length and the angle between them is increasing at a rate of $0.07$ rad/s. Find the rate at which the area of the triangle is increasing when the
Angle between the sides of fixed length is ($\frac{\pi}{3}$ )

Accepted Answer

A)  $-0.955 \mathrm{~m}^{2} / \mathrm{s}$ 
B)  $1.145 \mathrm{~m}^{2} / \mathrm{s}$ 
C)  $-1.955 \mathrm{~m}^{2} / \mathrm{s}$ 
D)  $0.105 \mathrm{~m}^{2} / \mathrm{s}$ 
E)  $5.045 \mathrm{~m}^{2} / \mathrm{s}$ 
A)  $-0.955 \mathrm{~m}^{2} / \mathrm{s}$ 
B)  $1.145 \mathrm{~m}^{2} / \mathrm{s}$ 
C)  $-1.955 \mathrm{~m}^{2} / \mathrm{s}$ 
D)  $0.105 \mathrm{~m}^{2} / \mathrm{s}$ 
E)  $5.045 \mathrm{~m}^{2} / \mathrm{s}$

Question 18

s(t) is the position of a body moving along a coordinate line; s(t) is measured in feet and t in seconds, where $t \geq 0$ . Find the position, velocity, and speed of the body at the indicated time. $s(t)=\frac{4 t}{t^{2}+1}$ ; t = 3

Accepted Answer

The answer of s(t) is the position of a body...

Question 19

Find an equation of the tangent line to the curve $90\left(x^{2}+y^{2}\right)^{2}=1734\left(x^{2}-y^{2}\right)$ at the point (4,1).

Accepted Answer

A)  $y=-1.11 x+17$ 
B)  $y=1.11 x+5.43$ 
C)  $y=-1.11 x+5.43$ 
D)  $y=-1.11 x+3.43$ 
E)  $\text { None of these }$ 
A)  $y=-1.11 x+17$ 
B)  $y=1.11 x+5.43$ 
C)  $y=-1.11 x+5.43$ 
D)  $y=-1.11 x+3.43$ 
E)  $\text { None of these }$

Question 20

Find an equation of the tangent line to the given curve at the indicated point. $\frac{1}{13} y^{2}-x^{3}-x^{2}=0 ; \quad(1, \sqrt{26})$

Accepted Answer

The answer of Find an equation of the tangent line...

Find the derivative of the function. $f(x)=2 \sqrt{x}+4 e^{x}$

Find the derivative of $f(x)$ . $f(x)=x^{8} \cosh x$

Find the second derivative of the function. $f(x)=5 e^{x} \cos x$

The function $f(t)=\frac{t}{t-8}$ satisfies the hypotheses of the Mean Value Theorem on the interval $[-2,0]$ . Find all values of c that satisfy the conclusion of the theorem.

Find an equation of the tangent line to the curve $90\left(x^{2}+y^{2}\right)^{2}=1734\left(x^{2}-y^{2}\right)$ at the point (4,1).

Find the derivative of the function. $f(u)=e^{u} \cot u$

Determine the values of x for which the given linear approximation is accurate to within 0.07 at a = 0. $\tan x \approx X$

Find an equation of the tangent line to the curve. $y=\frac{\sqrt{x}}{x+6} \text { at }(4,0.2)$

Find dy/dx by implicit differentiation. $5 x^{2}+9 y^{2}=5$

The altitude of a triangle is increasing at a rate of $1 \mathrm{~cm} / \min$ while the area of the triangle is increasing at a rate of $2 \mathrm{~cm}^{2} / \min$ . At what rate is the base of the triangle changing when the altitude is 10 cm and the area is $200 \mathrm{~cm}^{2}$ .

Find an equation of the tangent line to the curve $x e^{y}+x+2 y=2$ at (1, 0).

Calculate $y^{\prime}$ . $x y^{4}+x^{2} y=5 x+3 y$

Differentiate the function. $B(y)=c y^{-4}$

Find the first and the second derivatives of the function. $y=\frac{x}{5-x}$

Find the derivative of the function. $f(x)=-x^{2}+7 x+5$

Two sides of a triangle are 2 m and 3 m in length and the angle between them is increasing at a rate of $0.07$ rad/s. Find the rate at which the area of the triangle is increasing when the Angle between the sides of fixed length is ( $\frac{\pi}{3}$ )

s(t) is the position of a body moving along a coordinate line; s(t) is measured in feet and t in seconds, where $t \geq 0$ . Find the position, velocity, and speed of the body at the indicated time. $s(t)=\frac{4 t}{t^{2}+1}$ ; t = 3

Find an equation of the tangent line to the curve $90\left(x^{2}+y^{2}\right)^{2}=1734\left(x^{2}-y^{2}\right)$ at the point (4,1).

Find an equation of the tangent line to the given curve at the indicated point. $\frac{1}{13} y^{2}-x^{3}-x^{2}=0 ; \quad(1, \sqrt{26})$ $Find an equation of the tangent line to the given curve at the indicated point. \frac{1}{13} y^{2}-x^{3}-x^{2}=0 ; \quad(1, \sqrt{26})$

Functions and Limits

Derivatives

Integrals

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 3: Applications of Differentiation

Find the derivative of the function. f(x)=2x+4exf(x)=2 \sqrt{x}+4 e^{x}f(x)=2x​+4ex

Find the derivative of f(x)f(x)f(x) . f(x)=x8cosh⁡xf(x)=x^{8} \cosh xf(x)=x8coshx

Find the second derivative of the function. f(x)=5excos⁡xf(x)=5 e^{x} \cos xf(x)=5excosx

The function f(t)=tt−8f(t)=\frac{t}{t-8}f(t)=t−8t​ satisfies the hypotheses of the Mean Value Theorem on the interval [−2,0][-2,0][−2,0] . Find all values of c that satisfy the conclusion of the theorem.

Find an equation of the tangent line to the curve 90(x2+y2)2=1734(x2−y2)90\left(x^{2}+y^{2}\right)^{2}=1734\left(x^{2}-y^{2}\right)90(x2+y2)2=1734(x2−y2) at the point (4,1).

Find the derivative of the function. f(u)=eucot⁡uf(u)=e^{u} \cot uf(u)=eucotu

Determine the values of x for which the given linear approximation is accurate to within 0.07 at a = 0. tan⁡x≈X\tan x \approx Xtanx≈X

Find an equation of the tangent line to the curve. y=xx+6 at (4,0.2)y=\frac{\sqrt{x}}{x+6} \text { at }(4,0.2)y=x+6x​​ at (4,0.2)

Find dy/dx by implicit differentiation. 5x2+9y2=55 x^{2}+9 y^{2}=55x2+9y2=5

Find an equation of the tangent line to the curve xey+x+2y=2x e^{y}+x+2 y=2xey+x+2y=2 at (1, 0).

Calculate y′y^{\prime}y′ . xy4+x2y=5x+3yx y^{4}+x^{2} y=5 x+3 yxy4+x2y=5x+3y

Differentiate the function. B(y)=cy−4B(y)=c y^{-4}B(y)=cy−4

Find the first and the second derivatives of the function. y=x5−xy=\frac{x}{5-x}y=5−xx​

Find the derivative of the function. f(x)=−x2+7x+5f(x)=-x^{2}+7 x+5f(x)=−x2+7x+5

Two sides of a triangle are 2 m and 3 m in length and the angle between them is increasing at a rate of 0.070.070.07 rad/s. Find the rate at which the area of the triangle is increasing when the Angle between the sides of fixed length is ( π3\frac{\pi}{3}3π​ )

s(t) is the position of a body moving along a coordinate line; s(t) is measured in feet and t in seconds, where t≥0t \geq 0t≥0 . Find the position, velocity, and speed of the body at the indicated time. s(t)=4tt2+1s(t)=\frac{4 t}{t^{2}+1}s(t)=t2+14t​ ; t = 3

Find an equation of the tangent line to the curve 90(x2+y2)2=1734(x2−y2)90\left(x^{2}+y^{2}\right)^{2}=1734\left(x^{2}-y^{2}\right)90(x2+y2)2=1734(x2−y2) at the point (4,1).

Find an equation of the tangent line to the given curve at the indicated point. 113y2−x3−x2=0;(1,26)\frac{1}{13} y^{2}-x^{3}-x^{2}=0 ; \quad(1, \sqrt{26})131​y2−x3−x2=0;(1,26​)

Functions and Limits

Derivatives

Integrals

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 derivative of the function. $f(x)=2 \sqrt{x}+4 e^{x}$

Find the derivative of $f(x)$ . $f(x)=x^{8} \cosh x$

Find the second derivative of the function. $f(x)=5 e^{x} \cos x$

The function $f(t)=\frac{t}{t-8}$ satisfies the hypotheses of the Mean Value Theorem on the interval $[-2,0]$ . Find all values of c that satisfy the conclusion of the theorem.

Find an equation of the tangent line to the curve $90\left(x^{2}+y^{2}\right)^{2}=1734\left(x^{2}-y^{2}\right)$ at the point (4,1).

Find the derivative of the function. $f(u)=e^{u} \cot u$

Determine the values of x for which the given linear approximation is accurate to within 0.07 at a = 0. $\tan x \approx X$

Find an equation of the tangent line to the curve. $y=\frac{\sqrt{x}}{x+6} \text { at }(4,0.2)$

Find dy/dx by implicit differentiation. $5 x^{2}+9 y^{2}=5$

Find an equation of the tangent line to the curve $x e^{y}+x+2 y=2$ at (1, 0).

Calculate $y^{\prime}$ . $x y^{4}+x^{2} y=5 x+3 y$

Differentiate the function. $B(y)=c y^{-4}$

Find the first and the second derivatives of the function. $y=\frac{x}{5-x}$

Find the derivative of the function. $f(x)=-x^{2}+7 x+5$

Two sides of a triangle are 2 m and 3 m in length and the angle between them is increasing at a rate of $0.07$ rad/s. Find the rate at which the area of the triangle is increasing when the Angle between the sides of fixed length is ( $\frac{\pi}{3}$ )

s(t) is the position of a body moving along a coordinate line; s(t) is measured in feet and t in seconds, where $t \geq 0$ . Find the position, velocity, and speed of the body at the indicated time. $s(t)=\frac{4 t}{t^{2}+1}$ ; t = 3

Find an equation of the tangent line to the curve $90\left(x^{2}+y^{2}\right)^{2}=1734\left(x^{2}-y^{2}\right)$ at the point (4,1).

Find an equation of the tangent line to the given curve at the indicated point. $\frac{1}{13} y^{2}-x^{3}-x^{2}=0 ; \quad(1, \sqrt{26})$ $Find an equation of the tangent line to the given curve at the indicated point. \frac{1}{13} y^{2}-x^{3}-x^{2}=0 ; \quad(1, \sqrt{26})$