Back
Problem 4.7.75
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0⁺ x²ˣ
Problem 4.4.73
{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)
Problem 4.6.67
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
Problem 4.3.35
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x²/₃ (x²-4)
Problem 4.9.5
Give the antiderivatives of xᵖ . For what values of p does your answer apply?
Problem 4.3.40
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x² - 2 ln x
Problem 4.7.42
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)
Problem 4.9.19
Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.
ƒ(x) = eˣ
Problem 4.8.25
{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = ln x - x² + 3x - 1
Problem 4.9.15
Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.
p(x) = 3 sec² x
Problem 4.1.35
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = 1 / x + ln x
Problem 4.1.79
{Use of Tech} Absolute maxima and minima
a. Find the critical points of f on the given interval.
b. Determine the absolute extreme values of f on the given interval.
c. Use a graphing utility to confirm your conclusions.
f(x) = 2ᶻ sin x on [-2,6]
Problem 4.1.37
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = x² √(x + 5)
Problem 4.7.36
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (eˣ - x - 1) / 5x²
Problem 4.7.35
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→π (cos x +1 ) / (x - π )²
Problem 4.5.36
Covering a marble Imagine a flat-bottomed cylindrical pot with a circular cross section of radius 4. A marble with radius 0 < r < 4 is placed in the bottom of the pot. What is the radius of the marble that requires the most water to cover it completely?
Problem 4.5.25
Minimum distance Find the point P on the line y = 3x that is closest to the point (50, 0). What is the least distance between P and (50, 0)?
Problem 4.1.11
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>
Problem 4.7.32
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)
Problem 4.5.23
Closest point on a curve What point on the parabola y = 1 - x² is closest to the point (1, 1)?
Problem 4.3.83
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) = 2x³ - 3x² + 12
Problem 4.1.17
Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>
Problem 4.8.27
{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
Problem 4.9.103
A car starting at rest accelerates at 16 ft/s² for 5 seconds on a straight road. How far does it travel during this time?
Problem 4.7.46
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)
Problem 4.9.35
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((4x⁴ - 6x²) / x ) dx
Problem 4.1.25
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = x³ / 3 - 9x
Problem 4.8.48
{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) = x³ - 3x² + x + 1
Problem 4.4.51
49–54. {Use of Tech} Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points.
ƒ(x) = 3x⁴ + 4x³ - 12x²
Problem 4.6.57
Approximating changes
Approximate the change in the volume of a right circular cylinder of fixed radius r = 20 cm when its height decreases from h = 12 to h = 11.9 cm (V(h) = πr²h).