Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.

When a faucet is turned ​​on to fill a bathtub, the volume of water in gallons in the tub after t minutes is V(t)=3t. Find V′(12).

Sketch the graph of a function continuous on the given interval that satisfies the following conditions.

ƒ is continuous on the interval [-4, 4] ; f'(x) = 0 for x = -2, 0, and 3; ƒ has an absolute minimum at x = 3; ƒ has a local minimum at x = -2 ; ƒ has a local maximum at x = 0; ƒ has an absolute maximum at x = -4.

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

x⁴+y⁴ = 2;(1,−1)

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.

f(x) = 3x³ - 4x

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.

ƒ(x) = x³e⁻ˣ on [-1,5]

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.

ƒ(x) = x² - 10 on [-2, 3]

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 0⁺ (x - 3 √x) / (x - √x)

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.

ƒ(x) = sec x on [-(π/4),π/4]

{Use of Tech} Finding intersection points Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.

y = sin x and y = x/2

Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.

ƒ(x) = 2 sinx + 1

{Use of Tech} Tumor size In a study conducted at Dartmouth College, mice with a particular type of cancerous tumor were treated with the chemotherapy drug Cisplatin. If the volume of one of these tumors at the time of treatment is V₀, then the volume of the tumor t days after treatment is modeled by the function V(t) = V₀ (0.99e⁻⁰·¹²¹⁶ᵗ + 0.01e⁰·²³⁹ᵗ). (Source: Undergraduate Mathematics for the Life Sciences, MAA Notes No. 81, 2013)

Plot a grap​​h of y = 0.99e⁻⁰·¹²¹⁶ᵗ + 0.01e⁰·²³⁹ᵗ, for 0 ≤ t ≤ 16, and describe the tumor size over time. Use Newton’s method to determine when the tumor decreases to half of its original size.

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_z→0 (tan 4z) / (tan 7z)

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→∞ (ln(3x + 5eˣ)) / (ln(7x + 3e²ˣ)

{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.

x²/₃ + y²/₃ = 1 (Astroid or hypocycloid with four cusps)

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = 1 / x + ln x

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→∞ (tan⁻¹ x - π/2)/(1/x)

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.

ƒ(x) = sin 3x on [-π/4,π/3]

Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 0 (3 sin 4x) / 5x

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.

f(x) = x²e⁻ˣ

Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.

ƒ(x) = eˣ

{Use of Tech} Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations.

f(x) = tan x/2 on (-π,π)

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→∞ (log₂ x - log₃ x)

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.

ƒ(x) = x³ - 3x² on [-1, 3]

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.

f(x) = ln (1 - x)

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→0 (sin x - x) / 7x³

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.

ƒ(x) = (2x)ˣ on [0.1,1]

Linear approximation Find the linear approximation to the following functions at the given point a.

f(x) = 4x² + x; a = 1

Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

1/³√510