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mathematics
calculus 10th edition

Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions

In Exercises determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.y = x + sin x, 0 ≤ x < 2π
In Exercises find such that the line is tangent to the graph of the function. Function f(x) = kx³ Line y = x + 1
In Exercises use the alternative form of the derivative to find the derivative at x = c (if it exists). f(x) = x³ + 2x² + 1, c = -2
In Exercises determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. y = √3x + 2 cos x, 0 ≤ x < 2π
In Exercises find such that the line is tangent to the graph of the function. Function f(x) = kx4 Line y = 4x - 1
In Exercises find such that the line is tangent to the graph of the function. Function f(x) = k√√√x Line y = x + 4
In Exercises use the alternative form of the derivative to find the derivative at x = c (if it exists). g(x)=x²-x, c = 1
In Exercises use the alternative form of the derivative to find the derivative at x = c (if it exists). f(x) = x³ + 6x, c = 2
Use the graph ofƒ to answer each question. To print an enlarged copy of the graph, go to MathGraphs.com.(a) Between which two consecutive points is the average rate of change of the function greatest?(b) Is the average rate of change of the function between A and B greater than or less than the
In Exercises use the alternative form of the derivative to find the derivative at x = c (if it exists). g(x) = √√√√x, c = 0
In Exercises use the alternative form of the derivative to find the derivative at x = c (if it exists). g(x) = (x + 3)¹/³, c = −3 -3
In Exercises use the alternative form of the derivative to find the derivative at x = c (if it exists). f(x) = 3/x, c = 4
In Exercises use the alternative form of the derivative to find the derivative at x = c (if it exists). f(x) = |x6|, c = 6
In Exercises use the alternative form of the derivative to find the derivative at x = c (if it exists). f(x) = (x - 6)²/3, c = 6
In Exercises use the alternative form of the derivative to find the derivative at x = c (if it exists). h(x) = |x + 7, c = -7
Sketch the graph of a function ƒ such that ƒ' > 0 for all x and the rate of change of the function is decreasing.
In Exercises the graphs of a function ƒ and its derivative ƒ' are shown in the same set of coordinate axes. Label the graphs as ƒ or ƒ' and write a short paragraph stating the criteria you used in making your selection. To print an enlarged copy of the graph. 3. + -3-2-1 У -2- 1 2 3
In Exercises the graphs of a function ƒ and its derivative ƒ' are shown in the same set of coordinate axes. Label the graphs as ƒ or ƒ' and write a short paragraph stating the criteria you used in making your selection. To print an enlarged copy of the graph. 2 -2-1 y 2 3 4
In Exercises the relationship between ƒ and g is given. Explain the relationship between ƒ' and g'. g(x) = 2ƒ(x)
In Exercises describe the x-values at which ƒ is differentiable. f(x)=x²-9| + -4 -2 12 10 642 -4 y 2 4 X
In Exercises the relationship between ƒ and g is given. Explain the relationship between ƒ' and g'. g(x) = -5ƒ(x)
In Exercises the relationship between ƒ and g is given. Explain the relationship between ƒ' and g'. g(x) = 3ƒ(x) - 1
In Exercises describe the x-values at which ƒ is differentiable. f(x) = 4 2 -2 -4 y = 2 x - 3 2 4 6
In Exercises describe the x-values at which ƒ is differentiable. f(x) -4 x x2 – 4 5 4 3 2 134 X
In Exercises describe the x-values at which ƒ is differentiable. f(x) = √√x-1 3 درا 2 1 y 12 3 4
In Exercises describe the x-values at which ƒ is differentiable. f(x) = (x + 4)2/3 -6 -4 -2 4 -2 X
Show that the graphs of the two equationshave tangent lines that are perpendicular to each other at their point of intersection. y = x and y || 1 X
In Exercises describe the x-values at which ƒ is differentiable. f(x) -4 = √x² - 4₂ x ≤0 4- x², x > 0 y 40 2 4
In Exercises find an equation of the tangent line to the graph of the function ƒ through the point (x0, y0) not on the graph. To find the point of tangency (x, y) on the graph of ƒ, solve the equation f'(x) = Yo-y xo - x
In Exercises use a graphing utility to graph the function and find the x-values at which ƒ is differentiable. f(x) = |x - 5]
In Exercises find an equation of the tangent line to the graph of the function ƒ through the point (x0, y0) not on the graph. To find the point of tangency (x, y) on the graph of ƒ, solve the equation f'(x) = Yo-y xo - x
Sketch the graphs of y = x² and y = -x² + 6x - 5, and sketch the two lines that are tangent to both graphs. Find equations of these lines.
In Exercises use a graphing utility to graph the function and find the x-values at which ƒ is differentiable. f(x) = 4x x - 3
In Exercises use a graphing utility to graph the function and find the x-values at which ƒ is differentiable. f(x) = [x³ - 3x² + 3x, x² - 2x, (7) x ≤ 1 x > 1
Show that the graph of the function ƒ(x) = 3x + sin x + 2 does not have a horizontal tangent line.
Show that the graph of the functionƒ(x) = x5 + 3x³ + 5xdoes not have a tangent line with a slope of 3.
In Exercises find the derivatives from the left and from the right at x = 1 (if they exist). Is the function differentiable at x = 1? f(x) = √1-x²
In Exercises find the derivatives from the left and from the right at x = 1 (if they exist). Is the function differentiable at x = 1? f(x) = [(x - 1)³, [(x - 1)², x ≤ 1 x> 1
In Exercises find the derivatives from the left and from the right at x = 1 (if they exist). Is the function differentiable at x = 1? f(x) = X₂ x², x ≤ 1 x > 1
In Exercises , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f(x) 1 xn then f'(x) 1 ntn-1
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If ƒ(x) = g(x) + c, then ƒ'(x) = g'(x).
In Exercises find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. f(x) = sin x, EI 0, 6
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If y = π², then dy/dx = 2π.
LetandShow that ƒ is continuous, but not differentiable, at x = 0.Show that g is differentiable at 0, and find g'(0). f(x) = √x sin in / 0, x #0 x = 0
In Exercises the graph of a position function is shown. It represents the distance in miles that a person drives during a 10-minute trip to work. Make a sketch of the corresponding velocity function. Distance (in miles) S 10- 8- 6- 4- 2- (4,2) (10, 6) (6,2) |||| (0, 0) 2 4 6 8 10 Time (in minutes)
In Exercises the graph of a velocity function is shown. It represents the velocity in miles per hour during a 10-minute trip to work. Make a sketch of the corresponding position function. Velocity (in mi/h) 60 50 40 30 20 10 V [ 1 1 11 111 11 I 2 4 6 8 10 Time (in minutes)
In Exercises the graph of a position function is shown. It represents the distance in miles that a person drives during a 10-minute trip to work. Make a sketch of the corresponding velocity function. Distance (in miles) S H 10- 8- 6- 4- 2- (6,5), (10, 6) (8,5) (0, 0) 2 4 6 8 10 Time (in minutes)
In Exercises use the position function s(t) = -16t² + vot + so for free-falling objects.A ball is thrown straight down from the top of a 220-foot building with an initial velocity -22 of feet per second. What is its velocity after 3 seconds? What is its velocity after falling 108 feet?
A car is driven 15,000 miles a year and gets x miles per gallon. Assume that the average fuel cost is $3.48 per gallon. Find the annual cost of fuel C as a function of x and use this function to complete the table.Who would benefit more from a one-mile-per-gallon increase in fuel efficiency—the
In Exercises the graph of a velocity function is shown. It represents the velocity in miles per hour during a 10-minute trip to work. Make a sketch of the corresponding position function. Velocity (in mi/h) 60 50 40 30 20 10 V I I 2 4 6 8 10 Time (in minutes)
Use a graphing utility to graph the two functions ƒ(x) = x² + 1 and g(x)= x + 1 in the same viewing window.Use the zoom and trace features to analyze the graphs nearthe point (0, 1). What do you observe? Which function isdifferentiable at this point? Write a short paragraph describingthe
In Exercises use the position function s(t) = -4.9t² + vot + so for free-falling objects.A projectile is shot upward from the surface of Earth with an initial velocity of 120 meters per second. What is its velocity after 5 seconds? After 10 seconds?
In Exercises use the position function s(t) = -4.9t² + vot + so for free-falling objects.To estimate the height of a building, a stone is dropped from the top of the building into a pool of water at ground level. The splash is seen 5.6 seconds after the stone is dropped. What is the height of the
Verify that the average velocity over the time interval [to Δt, to + Δt] is the same as the instantaneous velocity at t = to for the position function s(t) = 1 -at2 + c. 2 रेवर
In Exercises find a and b such that ƒ is differentiable everywhere. f(x) = x ≤ 2 fax³, [x² + b₂ x > 2
The volume of a cube with sides of length s is given by V = s³. Find the rate of change of the volume with respect to s when s = 6 centimeters.
The area of a square with sides of length s is given by A = s². Find the rate of change of the area with respect to s when s = 6 meters.
Find an equation of the parabola y = ax² + bx + c that passes through (0, 1) and is tangent to the line y = x - 1 at (1, 0).
Let (a, b) be an arbitrary point on the graph of y = 1/x, x > 0. Prove that the area of the triangle formed by the tangent line through (a, b) and the coordinate axes is 2.
Find the equation(s) of the tangent line(s) to the graph of the curve y = x³ - 9x through the point (1, -9) not on the graph.
Find the equation(s) of the tangent line(s) to the graph of the parabola y = x² through the given point not on the graph.(a) (0, a) (b) (a, 0)Are there any restrictions on the constant a?
In Exercises find a and b such that ƒ is differentiable everywhere. f(x) = x < 0 [cos x, [ax + b₂ x ≥ 0
In Exercises complete the table to find the derivative of the function. Original Function y = πT (3x)² Rewrite Differentiate Simplify
In Exercises complete the table to find the derivative of the function. Original Function 6 y = (5r)3 Rewrite Differentiate Simplify
In Exercises complete the table to find the derivative of the function. Original Function Rewrite Differentiate Simplify 3 2x4 y
In Exercises identify a function ƒ that has the given characteristics. Then sketch the function. ƒ (0) = 4; ƒ'(0) = 0; f'(x) < 0 for x < 0; f'(x) > 0 for x > 0
In Exercises(a) Find an equation of the tangent line to the graph of ƒ atthe given point(b) Use a graphing utility to graph the functionand its tangent line at the point Function y = x² 3x² + 2 - Point (1, 0)
In Exercises find the derivative of the function. f(x) = ||N + 3 cos x
In Exercises identify a function ƒ that has the given characteristics. Then sketch the function. ƒ(0) = 2; ƒ'(x) = -3 for-∞o < x < ∞
In Exercises the limit represents ƒ'(c) for a function ƒ and a number c. Find ƒ and c. lim x-9 2√√√x-6 x - 9
In Exercises the limit represents ƒ'(c) for a function ƒ and a number c. Find ƒ and c. 9 - X 98 + zx- x-6 lim
In Exercises find the derivative of the function.ƒ(x) = 6√x + 5 cos x
In Exercises the limit represents ƒ'(c) for a function ƒ and a number c. Find ƒ and c. lim Δx-0 (−2 + Δx) + 8 Δε
In Exercises find the derivative of the function.ƒ(t) = t2/3 - t1/3 + 4
In Exercises find the derivative of the function. f(x)=√√√x - 63x
In Exercises sketch the graph of ƒ'. Explain how you found your answer. + -8 -4 fre -2 y 4 f foo 8 X
In Exercises the limit represents ƒ'(c) for a function ƒ and a number c. Find ƒ and c. lim Δx-0 [53(1 + Ax)] - 2 Δε
In Exercises find the derivative of the function. h(x) = 4x³ + 2x + 5 X
In Exercises find the derivative of the function. f(x) = x3 x³ - 3x² + 4 x²
In Exercises find the derivative of the function.y = x²(2x² - 3x)
The tangent line to the graph of y = h(x) at the point (-1, 4) passes through the point (3,6). Find h(-1) and h'(-1).
In Exercises find the derivative of the function.y = x(x² + 1)
In Exercises find the derivative of the function. f(x) = 2x4 는 X3
The tangent line to the graph of y = g(x) at the point (4, 5) passes through the point (7,0). Find g(4) and g'(4).
In Exercises sketch the graph of ƒ'. Explain how you found your answer. -3 -2 -1 4 3 y -2. f +x 1 2 3
In Exercises find the derivative of the function. f(x) = 4x³ + 3x² X
Sketch a graph of a function whose derivative is always positive. Explain how you found the answer.
Sketch a graph of a function whose derivative is always negative. Explain how you found the answer.
In Exercises find the derivative of the function. f(x) = 8x + 3 x²
In Exercises sketch the graph of ƒ'. Explain how you found your answer. 7 6 4 3 2 y + X 1 2 3 4 5 6 7 8
In Exercises find the derivative of the function. g(t) = 1² - 4 1³
In Exercises sketch the graph of ƒ'. Explain how you found your answer. 7 6 65432 4 3 T y f V + X 1 2 3 4 5 6 7
In Exercises sketch the graph of ƒ'. Explain how you found your answer. -4 -2 y -2. -6 2 f H +x 4
In Exercises find the derivative of the function.ƒ(x) = x³ - 2x + 3x-3
In Exercises sketch the graph of ƒ'. Explain how you found your answer. -3-2 3 2 y 11 -2 -3+ 1 2 3 -X
In Exercises find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Function g(t) = -2 cost + 5 Point (TT, 7)
In Exercises find an equation of the line that is tangent to the graph of and parallel to the given line. Function f(x) = 1 √√x Line x + 2y = 6 = 0
In Exercises find an equation of the line that is tangent to the graph of and parallel to the given line. Function f(x) = 1 √x x-1 Line x + 2y + 7 = 0
In Exercises find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Function f(x) = 2(x-4)² Point (2,8)
In Exercises find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Function f(0) = 4 sin 0 0 Point (0, 0)

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