Question 1

Find dy/dx by implicit differentiation. $3 \sqrt{x}+\sqrt{y}=6$

Accepted Answer

The answer of Find dy/dx by implicit differentiation. $3 \sqrt{x}+\sqrt{y}=6$...

Question 2

Find the derivative of the function. $f(x)=0.2 x^{-1.6}$

Accepted Answer

A)  $-0.32 x^{0.6}$ 
B)  $-\frac{0.32}{x^{0.6}}$ 
C)  $-\frac{0.32}{x^{2.6}}$ 
D)  $-0.32 x^{26}$ 
A)  $-0.32 x^{0.6}$ 
B)  $-\frac{0.32}{x^{0.6}}$ 
C)  $-\frac{0.32}{x^{2.6}}$ 
D)  $-0.32 x^{26}$

Question 3

Find an equation of the tangent line to the graph of the function at the indicated point. $f(x)=\frac{4}{x}$ $ \quad $$ \quad $$(4,1)$

Accepted Answer

A)  $y=4 x+2$ 
B)  $y=-4 x+2$ 
C)  $y=-\frac{1}{4} x+2$ 
D)  $y=\frac{1}{4} x+2$ 
A)  $y=4 x+2$ 
B)  $y=-4 x+2$ 
C)  $y=-\frac{1}{4} x+2$ 
D)  $y=\frac{1}{4} x+2$

Question 4

Find $d^{2} y / d x^{2}$ in terms of x and y. $x^{6}-y^{6}=-1$

Accepted Answer

A)  $30 x^{4}-30 y^{4}$ 
B)  $\frac{x^{5}}{y^{5}}$ 
C)  $6 x^{5}-6 y^{5}$ 
D)  $\frac{5 x^{4}}{y^{5}}-\frac{5 x^{10}}{y^{11}}$ 
A)  $30 x^{4}-30 y^{4}$ 
B)  $\frac{x^{5}}{y^{5}}$ 
C)  $6 x^{5}-6 y^{5}$ 
D)  $\frac{5 x^{4}}{y^{5}}-\frac{5 x^{10}}{y^{11}}$

Question 5

Find an equation of the line tangent to the graph of $y=\frac{e^{-9 x}}{x^{9}+1}$ at the point where x = 0.

Accepted Answer

A)  $y=-9 x+1$ 
B)  $y=9 x$ 
C)  $y=9 x+1$ 
D)  $y=-9 x$ 
A)  $y=-9 x+1$ 
B)  $y=9 x$ 
C)  $y=9 x+1$ 
D)  $y=-9 x$

Question 6

Find the limit. $\lim _{\theta \rightarrow 0} 4 \frac{\sin (\sin 4 \theta)}{\sec 4 \theta}$

Accepted Answer

The answer of Find the limit. \(\lim _{\theta \rightarrow 0}...

Question 7

Find the point(s) on the graph of f where the tangent line is horizontal. $f(x)=x^{8} e^{-x}$

Accepted Answer

A)  $\left(8, \frac{8^{8}}{e^{8}}\right)$ 
B)  (0, 0) 
C)  (0, 0), $\left(8, \frac{8^{8}}{e^{8}}\right)$ 
D)  $\left(1, \frac{1}{e}\right)$ 
A)  $\left(8, \frac{8^{8}}{e^{8}}\right)$ 
B)  (0, 0) 
C)  (0, 0), $\left(8, \frac{8^{8}}{e^{8}}\right)$ 
D)  $\left(1, \frac{1}{e}\right)$

Question 8

Use the table to estimate the value of $h^{\prime}(10.5) \text {, where } h(x)=f(g(x)) \text { and } g(10.5)=10.1$
$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline x & 10 & 10.1 & 10.2 & 10.3 & 10.4 & 10.5 & 10.6 \
\hline f^{\prime}(x) &4.5 & & 5.6 & 4.3 & 2.5 & 9.9 & 7.8 \
& & 3.3 & & & & & \
\hline g^{\prime}(x) & 6.5 & 5.9 & 4.7 & 4.2 & 5.4 & & 6.3 \
& & & & & &10 & \
\hline
\end{array}$

Accepted Answer

The answer of Use the table to estimate the value...

Question 9

Calculate y'. $y=4 \ln \left(x^{2} e^{x}\right)$

Accepted Answer

A)  $y^{\prime}=4\left(\frac{2}{x}+1\right)$ 
B)  $y^{\prime}=x \ln 2 x+x^{2}$ 
C)  $y^{\prime}=4\left(\frac{2}{x}+2\right)$ 
D)  $y^{\prime}=\frac{2 e^{x}+x \ln x^{2}}{x}$ 
E)  $y^{\prime}=3 x^{2}$ 
A)  $y^{\prime}=4\left(\frac{2}{x}+1\right)$ 
B)  $y^{\prime}=x \ln 2 x+x^{2}$ 
C)  $y^{\prime}=4\left(\frac{2}{x}+2\right)$ 
D)  $y^{\prime}=\frac{2 e^{x}+x \ln x^{2}}{x}$ 
E)  $y^{\prime}=3 x^{2}$

Question 10

The curve with the equation $x^{2 / 3}+y^{2 / 3}=4$ is called an asteroid. Find an equation of the tangent to the curve at the point ( $3 \sqrt{3}$ , 1).

Accepted Answer

The answer of The curve with the equation \(x^{2 /...

Question 11

A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is $x(t)=9 \sin t$ , where t is in seconds and x in centimeters. Find the velocity at time t.

Accepted Answer

A)  $v(t)=9 \cos t$ 
B)  $v(t)=2 \cos 9 t$ 
C)  $v(t)=\cos 9 t$ 
D)  $v(t)=9 \sin 9 t$ 
E)  $v(t)=\sin 9 t$ 
A)  $v(t)=9 \cos t$ 
B)  $v(t)=2 \cos 9 t$ 
C)  $v(t)=\cos 9 t$ 
D)  $v(t)=9 \sin 9 t$ 
E)  $v(t)=\sin 9 t$

Question 12

Find $f^{\prime}$ in terms of $g^{\prime}$ . $f(x)=x^{8} g(x)$

Accepted Answer

The answer of Find $f^{\prime}$ in terms of $g^{\prime}$ ....

Question 13

Find the derivative of the function.
$y=2 \cos ^{-1}\left(\sin ^{-1} t\right)$

Accepted Answer

The answer of Find the derivative of the function.
\(y=2 \cos...

Question 14

Find an equation of the tangent line to the curve. $y=\frac{\sqrt{x}}{x+6} \text { at }(4,0.6)$

Accepted Answer

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

Question 15

A telephone line hangs between two poles at 12 m apart in the shape of the catenary $y=30 \cosh \left(\frac{x}{30}\right)-35$ , where x and y are measured in meters. Find the slope of this curve where it meets the right pole.

Accepted Answer

A)  $\frac{\sinh 5}{6}$ 
B)  $\sinh \left(\frac{5}{6}\right)$ 
C)  $\sinh \left(\frac{1}{6}\right)$ 
D)  $\sinh 6$ 
E)  $\sinh \left(\frac{1}{5}\right)$ 
A)  $\frac{\sinh 5}{6}$ 
B)  $\sinh \left(\frac{5}{6}\right)$ 
C)  $\sinh \left(\frac{1}{6}\right)$ 
D)  $\sinh 6$ 
E)  $\sinh \left(\frac{1}{5}\right)$

Question 16

Find $f^{\prime}$ in terms of $g^{\prime}$ . $f(x)=[g(x)]^{5}$

Accepted Answer

A)  $f^{\prime}(x)=5[g x]\left[x g^{\prime}+g\right]$ 
B)  $f^{\prime}(x)=5 g(x)$ 
C)  $f^{\prime}(x)=5 g^{\prime}(x)$ 
D)  $f^{\prime}(x)=5\left[g^{\prime}(x)\right]^{4}$ 
E)  $f^{\prime}(x)=5[g(x)]^{4} g^{\prime}(x)$ 
A)  $f^{\prime}(x)=5[g x]\left[x g^{\prime}+g\right]$ 
B)  $f^{\prime}(x)=5 g(x)$ 
C)  $f^{\prime}(x)=5 g^{\prime}(x)$ 
D)  $f^{\prime}(x)=5\left[g^{\prime}(x)\right]^{4}$ 
E)  $f^{\prime}(x)=5[g(x)]^{4} g^{\prime}(x)$

Question 17

Find the derivative of the function. $f(x)=\left(4 x+5 e^{x}\right)\left(x-e^{x}\right)$

Accepted Answer

The answer of Find the derivative of the function. \(f(x)=\left(4...

Question 18

The parents of a child wish to establish a trust fund for the child's college education. If they need an estimated $90,000 5 years from now and they are able to invest the money at 5.5% compounded continuously in the interim, how much should they set aside in trust now?

Accepted Answer

A)  $68,361.49 
B)  $17,061.61 
C)  $17,036.73 
D)  $68,862.09 
A)  $68,361.49 
B)  $17,061.61 
C)  $17,036.73 
D)  $68,862.09

Question 19

Find the derivative of the function. $f(t)$ = cosh2 (6t2 + 3)

Accepted Answer

A)  24t sinh (6t2 + 3) 
B)  24t cosh (6t2 + 3) sinh (6t2 + 3) 
C)  12t sinh (6t2 + 3) 
D)  12t cosh (6t2 + 3) sinh (6t2 + 3) 
A)  24t sinh (6t2 + 3) 
B)  24t cosh (6t2 + 3) sinh (6t2 + 3) 
C)  12t sinh (6t2 + 3) 
D)  12t cosh (6t2 + 3) sinh (6t2 + 3)

Question 20

Suppose that f and g are functions that are differentiable at x = 2 and that f (2) = -1, $f^{\prime}$ (2) = 3, g(2) = 3, and $g^{\prime}$ (2) = -4. Find $h^{\prime}(2)$ . $h(x)=\frac{x f(x)}{x+g(x)}$

Accepted Answer

The answer of Suppose that f and g are functions...

Find dy/dx by implicit differentiation. $3 \sqrt{x}+\sqrt{y}=6$

Find the derivative of the function. $f(x)=0.2 x^{-1.6}$

Find an equation of the tangent line to the graph of the function at the indicated point. $f(x)=\frac{4}{x}$ $\quad$ $\quad$ $(4,1)$

Find $d^{2} y / d x^{2}$ in terms of x and y. $x^{6}-y^{6}=-1$

Find an equation of the line tangent to the graph of $y=\frac{e^{-9 x}}{x^{9}+1}$ at the point where x = 0.

Find the limit. $\lim _{\theta \rightarrow 0} 4 \frac{\sin (\sin 4 \theta)}{\sec 4 \theta}$

Find the point(s) on the graph of f where the tangent line is horizontal. $f(x)=x^{8} e^{-x}$

Use the table to estimate the value of $h^{\prime}(10.5) \text {, where } h(x)=f(g(x)) \text { and } g(10.5)=10.1$ x 10 10.1 10.2 10.3 10.4 10.5 10.6 (x) 4.5 5.6 4.3 2.5 9.9 7.8 3.3 (x) 6.5 5.9 4.7 4.2 5.4 6.3 10

Calculate y'. $y=4 \ln \left(x^{2} e^{x}\right)$

Find $f^{\prime}$ in terms of $g^{\prime}$ . $f(x)=x^{8} g(x)$

Find the derivative of the function. $y=2 \cos ^{-1}\left(\sin ^{-1} t\right)$

Find an equation of the tangent line to the curve. $y=\frac{\sqrt{x}}{x+6} \text { at }(4,0.6)$

Find $f^{\prime}$ in terms of $g^{\prime}$ . $f(x)=[g(x)]^{5}$

Find the derivative of the function. $f(x)=\left(4 x+5 e^{x}\right)\left(x-e^{x}\right)$

The parents of a child wish to establish a trust fund for the child's college education. If they need an estimated $90,000 5 years from now and they are able to invest the money at 5.5% compounded continuously in the interim, how much should they set aside in trust now?

Find the derivative of the function. $f(t)$ = cosh²(6t² + 3)

Suppose that f and g are functions that are differentiable at x = 2 and that f (2) = -1, $f^{\prime}$ (2) = 3, g(2) = 3, and $g^{\prime}$ (2) = -4. Find $h^{\prime}(2)$ . $h(x)=\frac{x f(x)}{x+g(x)}$

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 dy/dx by implicit differentiation. 3x+y=63 \sqrt{x}+\sqrt{y}=63x​+y​=6

Find the derivative of the function. f(x)=0.2x−1.6f(x)=0.2 x^{-1.6}f(x)=0.2x−1.6

Find an equation of the tangent line to the graph of the function at the indicated point. f(x)=4xf(x)=\frac{4}{x}f(x)=x4​ \quad \quad (4,1)(4,1)(4,1)

Find d2y/dx2d^{2} y / d x^{2}d2y/dx2 in terms of x and y. x6−y6=−1x^{6}-y^{6}=-1x6−y6=−1

Find an equation of the line tangent to the graph of y=e−9xx9+1y=\frac{e^{-9 x}}{x^{9}+1}y=x9+1e−9x​ at the point where x = 0.

Find the limit. lim⁡θ→04sin⁡(sin⁡4θ)sec⁡4θ\lim _{\theta \rightarrow 0} 4 \frac{\sin (\sin 4 \theta)}{\sec 4 \theta}limθ→0​4sec4θsin(sin4θ)​

Find the point(s) on the graph of f where the tangent line is horizontal. f(x)=x8e−xf(x)=x^{8} e^{-x}f(x)=x8e−x

Calculate y'. y=4ln⁡(x2ex)y=4 \ln \left(x^{2} e^{x}\right)y=4ln(x2ex)

The curve with the equation x2/3+y2/3=4x^{2 / 3}+y^{2 / 3}=4x2/3+y2/3=4 is called an asteroid. Find an equation of the tangent to the curve at the point ( 333 \sqrt{3}33​ , 1).

A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is x(t)=9sin⁡tx(t)=9 \sin tx(t)=9sint , where t is in seconds and x in centimeters. Find the velocity at time t.

Find f′f^{\prime}f′ in terms of g′g^{\prime}g′ . f(x)=x8g(x)f(x)=x^{8} g(x)f(x)=x8g(x)

Find the derivative of the function. y=2cos⁡−1(sin⁡−1t)y=2 \cos ^{-1}\left(\sin ^{-1} t\right)y=2cos−1(sin−1t)

Find an equation of the tangent line to the curve. y=xx+6 at (4,0.6)y=\frac{\sqrt{x}}{x+6} \text { at }(4,0.6)y=x+6x​​ at (4,0.6)

A telephone line hangs between two poles at 12 m apart in the shape of the catenary y=30cosh⁡(x30)−35y=30 \cosh \left(\frac{x}{30}\right)-35y=30cosh(30x​)−35 , where x and y are measured in meters. Find the slope of this curve where it meets the right pole.

Find f′f^{\prime}f′ in terms of g′g^{\prime}g′ . f(x)=[g(x)]5f(x)=[g(x)]^{5}f(x)=[g(x)]5

Find the derivative of the function. f(x)=(4x+5ex)(x−ex)f(x)=\left(4 x+5 e^{x}\right)\left(x-e^{x}\right)f(x)=(4x+5ex)(x−ex)

The parents of a child wish to establish a trust fund for the child's college education. If they need an estimated $90,000 5 years from now and they are able to invest the money at 5.5% compounded continuously in the interim, how much should they set aside in trust now?

Find the derivative of the function. f(t)f(t)f(t) = cosh2 (6t2 + 3)

Suppose that f and g are functions that are differentiable at x = 2 and that f (2) = -1, f′f^{\prime}f′ (2) = 3, g(2) = 3, and g′g^{\prime}g′ (2) = -4. Find h′(2)h^{\prime}(2)h′(2) . h(x)=xf(x)x+g(x)h(x)=\frac{x f(x)}{x+g(x)}h(x)=x+g(x)xf(x)​

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 dy/dx by implicit differentiation. $3 \sqrt{x}+\sqrt{y}=6$

Find the derivative of the function. $f(x)=0.2 x^{-1.6}$

Find an equation of the tangent line to the graph of the function at the indicated point. $f(x)=\frac{4}{x}$ $\quad$ $\quad$ $(4,1)$

Find $d^{2} y / d x^{2}$ in terms of x and y. $x^{6}-y^{6}=-1$

Find an equation of the line tangent to the graph of $y=\frac{e^{-9 x}}{x^{9}+1}$ at the point where x = 0.

Find the limit. $\lim _{\theta \rightarrow 0} 4 \frac{\sin (\sin 4 \theta)}{\sec 4 \theta}$

Find the point(s) on the graph of f where the tangent line is horizontal. $f(x)=x^{8} e^{-x}$

Calculate y'. $y=4 \ln \left(x^{2} e^{x}\right)$

Find $f^{\prime}$ in terms of $g^{\prime}$ . $f(x)=x^{8} g(x)$

Find the derivative of the function. $y=2 \cos ^{-1}\left(\sin ^{-1} t\right)$

Find an equation of the tangent line to the curve. $y=\frac{\sqrt{x}}{x+6} \text { at }(4,0.6)$

Find $f^{\prime}$ in terms of $g^{\prime}$ . $f(x)=[g(x)]^{5}$

Find the derivative of the function. $f(x)=\left(4 x+5 e^{x}\right)\left(x-e^{x}\right)$

Find the derivative of the function. $f(t)$ = cosh²(6t² + 3)

Suppose that f and g are functions that are differentiable at x = 2 and that f (2) = -1, $f^{\prime}$ (2) = 3, g(2) = 3, and $g^{\prime}$ (2) = -4. Find $h^{\prime}(2)$ . $h(x)=\frac{x f(x)}{x+g(x)}$