Exam 4: Integrals

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The quantity demanded per month of an item is related to the unit price by the demand equation $p=\frac{40}{0.05 x^{2}+5}$ , $0 \leq x \leq 20$ where p is measured in dollars and x is measured in units of a thousand. How many items must be sold by the manufacturer to maximize its revenue? Hint: Recall that the revenue is given by $R=p x$

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Question 161

Use l'Hospital's Rule to calculate the exact value of the limit $\frac{f(x)}{g(x)}$ as $x \rightarrow 0$ . $f(x)=e^{x}-4$ and $g(x)=x^{5}+6 x$

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Question 162

For the function f whose graph is given, find the limit. $\lim _{x \rightarrow-1^{-}} f(x)$ $For the function f whose graph is given, find the limit. \lim _{x \rightarrow-1^{-}} f(x)$

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Question 163

Use Newton's method to approximate the indicated root of $x^{4}+x-4=0$ in the interval $[1,2]$ , correct to six decimal places. Use $x_{1}=1.5$ as the initial approximation.

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Question 164

Find the position function of a particle moving along a coordinate line that satisfies the given conditions. $a(t)=4 \sin t-\cos t$ , s (0) = 5, v (0) = 0

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Question 165

The function $f(x)=\frac{1}{3} x^{3}-16 x$ satisfies the hypotheses of Rolle's Theorem on the interval $[0,4 \sqrt{3}]$ . Find all values of c that satisfy the conclusion of the theorem.

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Question 166

A farmer with 710 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens?

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Question 167

The graph of the first derivative $f^{\prime}(x)$ of a function f is shown below. At what values of x does f have a local maximum or minimum? $The graph of the first derivative f^{\prime}(x) of a function f is shown below. At what values of x does f have a local maximum or minimum?$

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Question 168

Given $f(x)=x+\frac{1}{x}$ . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

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Question 169

Showing 161 - 169 of 169

Use l'Hospital's Rule to calculate the exact value of the limit $\frac{f(x)}{g(x)}$ as $x \rightarrow 0$ . $f(x)=e^{x}-4$ and $g(x)=x^{5}+6 x$

For the function f whose graph is given, find the limit. $\lim _{x \rightarrow-1^{-}} f(x)$ $For the function f whose graph is given, find the limit. \lim _{x \rightarrow-1^{-}} f(x)$

Use Newton's method to approximate the indicated root of $x^{4}+x-4=0$ in the interval $[1,2]$ , correct to six decimal places. Use $x_{1}=1.5$ as the initial approximation.

Find the position function of a particle moving along a coordinate line that satisfies the given conditions. $a(t)=4 \sin t-\cos t$ , s (0) = 5, v (0) = 0

The function $f(x)=\frac{1}{3} x^{3}-16 x$ satisfies the hypotheses of Rolle's Theorem on the interval $[0,4 \sqrt{3}]$ . Find all values of c that satisfy the conclusion of the theorem.

A farmer with 710 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens?

The graph of the first derivative $f^{\prime}(x)$ of a function f is shown below. At what values of x does f have a local maximum or minimum? $The graph of the first derivative f^{\prime}(x) of a function f is shown below. At what values of x does f have a local maximum or minimum?$

Given $f(x)=x+\frac{1}{x}$ . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

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

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Exam 4: Integrals

Use l'Hospital's Rule to calculate the exact value of the limit f(x)g(x)\frac{f(x)}{g(x)}g(x)f(x)​ as x→0x \rightarrow 0x→0 . f(x)=ex−4f(x)=e^{x}-4f(x)=ex−4 and g(x)=x5+6xg(x)=x^{5}+6 xg(x)=x5+6x

For the function f whose graph is given, find the limit. lim⁡x→−1−f(x)\lim _{x \rightarrow-1^{-}} f(x)limx→−1−​f(x)

Use Newton's method to approximate the indicated root of x4+x−4=0x^{4}+x-4=0x4+x−4=0 in the interval [1,2][1,2][1,2] , correct to six decimal places. Use x1=1.5x_{1}=1.5x1​=1.5 as the initial approximation.

Find the position function of a particle moving along a coordinate line that satisfies the given conditions. a(t)=4sin⁡t−cos⁡ta(t)=4 \sin t-\cos ta(t)=4sint−cost , s (0) = 5, v (0) = 0

The function f(x)=13x3−16xf(x)=\frac{1}{3} x^{3}-16 xf(x)=31​x3−16x satisfies the hypotheses of Rolle's Theorem on the interval [0,43][0,4 \sqrt{3}][0,43​] . Find all values of c that satisfy the conclusion of the theorem.

A farmer with 710 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens?

The graph of the first derivative f′(x)f^{\prime}(x)f′(x) of a function f is shown below. At what values of x does f have a local maximum or minimum?

Given f(x)=x+1xf(x)=x+\frac{1}{x}f(x)=x+x1​ . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

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

Use l'Hospital's Rule to calculate the exact value of the limit $\frac{f(x)}{g(x)}$ as $x \rightarrow 0$ . $f(x)=e^{x}-4$ and $g(x)=x^{5}+6 x$

For the function f whose graph is given, find the limit. $\lim _{x \rightarrow-1^{-}} f(x)$ $For the function f whose graph is given, find the limit. \lim _{x \rightarrow-1^{-}} f(x)$

Use Newton's method to approximate the indicated root of $x^{4}+x-4=0$ in the interval $[1,2]$ , correct to six decimal places. Use $x_{1}=1.5$ as the initial approximation.

Find the position function of a particle moving along a coordinate line that satisfies the given conditions. $a(t)=4 \sin t-\cos t$ , s (0) = 5, v (0) = 0

The function $f(x)=\frac{1}{3} x^{3}-16 x$ satisfies the hypotheses of Rolle's Theorem on the interval $[0,4 \sqrt{3}]$ . Find all values of c that satisfy the conclusion of the theorem.

The graph of the first derivative $f^{\prime}(x)$ of a function f is shown below. At what values of x does f have a local maximum or minimum? $The graph of the first derivative f^{\prime}(x) of a function f is shown below. At what values of x does f have a local maximum or minimum?$

Given $f(x)=x+\frac{1}{x}$ . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.