Exam 12: Applications of the Derivative

A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits only one-fourth as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit the most light. Neglect the thickness of the frame.

(Multiple Choice)

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Question 181

Find the equation of the tangent line at the given point on the curve. - $x^{2}+y^{2}+2 y=0 ;(0,-2)$

(Multiple Choice)

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Question 182

Find $\frac{d y}{d x}$ at the given point. - $x y^{2}=4 ;(4,1)$

(Multiple Choice)

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Question 183

Find the location and value of each local extremum for the function. -

(Multiple Choice)

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Question 184

Find the coordinates of the points of inflection for the function. - $f(x)=x^{3}+9 x^{2}+24 x+21$

(Multiple Choice)

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Question 185

Solve the problem. -Find the velocity function $v(t)$ if $s(t)=\frac{-3}{4 t+5}$ .

(Multiple Choice)

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Question 186

The rule of the derivative of a function $f$ is given. Find the location of all local extrema. - $f^{\prime}(x)=\left(x^{2}-1\right)(x-3)$

(Multiple Choice)

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Question 187

Find the location and value of each local extremum for the function. -

(Multiple Choice)

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Question 188

Solve the problem. -Find the acceleration function $a(t)$ if $s(t)=\frac{-3}{5 t+5}$ .

(Multiple Choice)

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Question 189

Evaluate $\mathrm{f}^{\prime \prime}(\mathrm{c})$ at the point. - $f(x)=\frac{3-x}{2 x+3}, c=-2$

(Multiple Choice)

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Question 190

The rule of the derivative of a function $f$ is given. Find the location of all local extrema. - $f^{\prime}(x)=\left(x^{2}-1\right)(x+3)$

(Multiple Choice)

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Question 191

Find the largest open interval where the function is changing as requested. -Increasing $y=7 x-5$

(Multiple Choice)

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Question 192

Use the first derivative test to determine the location of each local extremum and the value of the function at thatextremum. - $f(x)=\frac{x^{2}}{x^{2}+5}$

(Multiple Choice)

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Question 193

Sketch the graph and show all local extrema and inflection points. - $f(x)=2 x^{3}+3 x^{2}-12 x$ $Sketch the graph and show all local extrema and inflection points. - f(x)=2 x^{3}+3 x^{2}-12 x$

(Multiple Choice)

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Question 194

Find the largest open intervals where the function is concave upward. - $f(x)=x^{4}-6 x^{3}$

(Multiple Choice)

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Question 195

Find the location of the indicated absolute extrema for the function. -Minimum

(Multiple Choice)

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Question 196

Find the location and value of each local extremum for the function. -

(Multiple Choice)

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Question 197

Solve the problem. -A container, in the shape of an inverted right circular cone, has a radius of 4 inches at the top and a height of 8 inches. At the instant when the water in the container is 5 inches deep, the surface level is falling at the rate of $-1 \mathrm{in} . / \mathrm{s}$ . Find the rate at which water is being drained.

(Multiple Choice)

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Question 198

Use calculus and a graphing calculator to find the approximate location of all relative extrema. - $f(x)=x^{4}-4 x^{3}-53 x^{2}-86 x-43$

(Multiple Choice)

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Question 199

Find dy/dx by implicit differentiation. - $x^{1 / 3}-y^{1 / 3}=1$

(Multiple Choice)

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Question 200

Find the equation of the tangent line at the given point on the curve. - $x^{2}+y^{2}+2 y=0 ;(0,-2)$

Find $\frac{d y}{d x}$ at the given point. - $x y^{2}=4 ;(4,1)$

Find the location and value of each local extremum for the function. -

Find the coordinates of the points of inflection for the function. - $f(x)=x^{3}+9 x^{2}+24 x+21$

Solve the problem. -Find the velocity function $v(t)$ if $s(t)=\frac{-3}{4 t+5}$ .

The rule of the derivative of a function $f$ is given. Find the location of all local extrema. - $f^{\prime}(x)=\left(x^{2}-1\right)(x-3)$

Find the location and value of each local extremum for the function. -

Solve the problem. -Find the acceleration function $a(t)$ if $s(t)=\frac{-3}{5 t+5}$ .

Evaluate $\mathrm{f}^{\prime \prime}(\mathrm{c})$ at the point. - $f(x)=\frac{3-x}{2 x+3}, c=-2$

The rule of the derivative of a function $f$ is given. Find the location of all local extrema. - $f^{\prime}(x)=\left(x^{2}-1\right)(x+3)$

Find the largest open interval where the function is changing as requested. -Increasing $y=7 x-5$

Use the first derivative test to determine the location of each local extremum and the value of the function at thatextremum. - $f(x)=\frac{x^{2}}{x^{2}+5}$

Sketch the graph and show all local extrema and inflection points. - $f(x)=2 x^{3}+3 x^{2}-12 x$ $Sketch the graph and show all local extrema and inflection points. - f(x)=2 x^{3}+3 x^{2}-12 x$

Find the largest open intervals where the function is concave upward. - $f(x)=x^{4}-6 x^{3}$

Find the location of the indicated absolute extrema for the function. -Minimum

Find the location and value of each local extremum for the function. -

Use calculus and a graphing calculator to find the approximate location of all relative extrema. - $f(x)=x^{4}-4 x^{3}-53 x^{2}-86 x-43$

Find dy/dx by implicit differentiation. - $x^{1 / 3}-y^{1 / 3}=1$

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Functions and Graphs

Exponential and Logarithmic Functions

Mathematics of Finance

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Introduction to Statistics

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Exam 12: Applications of the Derivative

Find the equation of the tangent line at the given point on the curve. - x2+y2+2y=0;(0,−2)x^{2}+y^{2}+2 y=0 ;(0,-2)x2+y2+2y=0;(0,−2)

Find dydx\frac{d y}{d x}dxdy​ at the given point. - xy2=4;(4,1)x y^{2}=4 ;(4,1)xy2=4;(4,1)

Find the location and value of each local extremum for the function. -

Find the coordinates of the points of inflection for the function. - f(x)=x3+9x2+24x+21f(x)=x^{3}+9 x^{2}+24 x+21f(x)=x3+9x2+24x+21

Solve the problem. -Find the velocity function v(t)v(t)v(t) if s(t)=−34t+5s(t)=\frac{-3}{4 t+5}s(t)=4t+5−3​ .

The rule of the derivative of a function fff is given. Find the location of all local extrema. - f′(x)=(x2−1)(x−3)f^{\prime}(x)=\left(x^{2}-1\right)(x-3)f′(x)=(x2−1)(x−3)

Find the location and value of each local extremum for the function. -

Solve the problem. -Find the acceleration function a(t)a(t)a(t) if s(t)=−35t+5s(t)=\frac{-3}{5 t+5}s(t)=5t+5−3​ .

Evaluate f′′(c)\mathrm{f}^{\prime \prime}(\mathrm{c})f′′(c) at the point. - f(x)=3−x2x+3,c=−2f(x)=\frac{3-x}{2 x+3}, c=-2f(x)=2x+33−x​,c=−2

The rule of the derivative of a function fff is given. Find the location of all local extrema. - f′(x)=(x2−1)(x+3)f^{\prime}(x)=\left(x^{2}-1\right)(x+3)f′(x)=(x2−1)(x+3)

Find the largest open interval where the function is changing as requested. -Increasing y=7x−5y=7 x-5y=7x−5

Use the first derivative test to determine the location of each local extremum and the value of the function at thatextremum. - f(x)=x2x2+5f(x)=\frac{x^{2}}{x^{2}+5}f(x)=x2+5x2​

Sketch the graph and show all local extrema and inflection points. - f(x)=2x3+3x2−12xf(x)=2 x^{3}+3 x^{2}-12 xf(x)=2x3+3x2−12x

Find the largest open intervals where the function is concave upward. - f(x)=x4−6x3f(x)=x^{4}-6 x^{3}f(x)=x4−6x3

Find the location of the indicated absolute extrema for the function. -Minimum

Find the location and value of each local extremum for the function. -

Use calculus and a graphing calculator to find the approximate location of all relative extrema. - f(x)=x4−4x3−53x2−86x−43f(x)=x^{4}-4 x^{3}-53 x^{2}-86 x-43f(x)=x4−4x3−53x2−86x−43

Find dy/dx by implicit differentiation. - x1/3−y1/3=1x^{1 / 3}-y^{1 / 3}=1x1/3−y1/3=1

Algebra and Equations

Graphs, Lines, and Inequalities

Functions and Graphs

Exponential and Logarithmic Functions

Mathematics of Finance

Systems of Linear Equations and Matrices

Linear Programming

Sets and Probability

Counting, Probability Distributions, and Further Topics in Probability

Introduction to Statistics

Differential Calculus

Integral Calculus

Multivariate Calculus

Filters

Find the equation of the tangent line at the given point on the curve. - $x^{2}+y^{2}+2 y=0 ;(0,-2)$

Find $\frac{d y}{d x}$ at the given point. - $x y^{2}=4 ;(4,1)$

Find the coordinates of the points of inflection for the function. - $f(x)=x^{3}+9 x^{2}+24 x+21$

Solve the problem. -Find the velocity function $v(t)$ if $s(t)=\frac{-3}{4 t+5}$ .

The rule of the derivative of a function $f$ is given. Find the location of all local extrema. - $f^{\prime}(x)=\left(x^{2}-1\right)(x-3)$

Solve the problem. -Find the acceleration function $a(t)$ if $s(t)=\frac{-3}{5 t+5}$ .

Evaluate $\mathrm{f}^{\prime \prime}(\mathrm{c})$ at the point. - $f(x)=\frac{3-x}{2 x+3}, c=-2$

The rule of the derivative of a function $f$ is given. Find the location of all local extrema. - $f^{\prime}(x)=\left(x^{2}-1\right)(x+3)$

Find the largest open interval where the function is changing as requested. -Increasing $y=7 x-5$

Use the first derivative test to determine the location of each local extremum and the value of the function at thatextremum. - $f(x)=\frac{x^{2}}{x^{2}+5}$

Sketch the graph and show all local extrema and inflection points. - $f(x)=2 x^{3}+3 x^{2}-12 x$ $Sketch the graph and show all local extrema and inflection points. - f(x)=2 x^{3}+3 x^{2}-12 x$

Find the largest open intervals where the function is concave upward. - $f(x)=x^{4}-6 x^{3}$

Use calculus and a graphing calculator to find the approximate location of all relative extrema. - $f(x)=x^{4}-4 x^{3}-53 x^{2}-86 x-43$

Find dy/dx by implicit differentiation. - $x^{1 / 3}-y^{1 / 3}=1$