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mathematics
calculus 10th edition
Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions
In Exercises find an equation of the tangent line to the graph at the given point. 8 4 (-2,-) y |f(x) = -8+ 4 16x x² + 16 + x 8
In Exercises(a) Find an equation of the tangent line to the graph off at the given point(b) Use a graphing utility to graph the function and its tangent line at the point y = (4x³ + 3)², (-1, 1)
In Exercises(a) Find an equation of the tangent line to the graph off at the given point(b) Use a graphing utility to graph the function and its tangent line at the point ƒ(x) = x√x² + 5, (2, 2)
In Exercises determine the point(s) at which the graph of the function has a horizontal tangent line. f(x) = 2x 1 x²
In Exercises find an equation of the tangent line to the graph at the given point. -4 -2 6 4 -2- f(x) = 8 2+4 2 X (2, 1) 4 X
Find equations of both tangent lines to the graph of the ellipse that pass through the point (4, 0) not on the graph. X 4 9 = 1
In Exercises find an equation of the tangent line to the graph at the given point. f(x) = 432 2 - 4x x² +6 y (2,3) 1 2 3 4 X
In Exercises(a) Find an equation of the tangent line to the graph of ƒ at the given point (b) Use a graphing utility to graph the function and its tangent line at the point (c) Use the derivative feature of a graphing utility to confirm your results. f(x) = tan x, 4
Prove (Theorem 2.3) thatfor the case in which n is a rational number. Write y = xp/q in the form yq = xp and differentiate implicitly. Assume that p and q are integers, where q > 0.) THEOREM 2.3 The Power Rule If n is a rational number, then the function f(x) =x" is differentiable
In Exercises find and evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. y = 26 sec ³ 4x, (0,25)
In Exercises find and evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. 5 /(x)=x²-²-₂ (-2-²) X 2' 3
The figure below shows the topographic map carried by a group of hikers. The hikers are in a wooded area on top of the hill shown on the map, and they decide to follow the path of steepest descent (orthogonal trajectories to the contours on the map). Draw their routes if they start from point A and
In Exercises find and evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. f(x) = x + 4 2x - 5' (9, 1)
In Exercises find an equation of the tangent line to the graph at the given point. -4 -2 6 4 -2 f(x) = 2 27 x² +9 4 X
In Exercises(a) Find an equation of the tangent line to the graph of ƒ at the given point (b) Use a graphing utility to graph the function and its tangent line at the point (c) Use the derivative feature of a graphing utility to confirm your results. f(x)= = X x + 4' (-5,5)
In Exercises find and evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. f(t) = 3t+2 t - 1' (0, -2)
Let L be any tangent line to the curveShow that the sum of the x- and y-intercepts of L is c √x + √y = √c.
In Exercises(a) Find an equation of the tangent line to the graph of ƒ at the given point (b) Use a graphing utility to graph the function and its tangent line at the point (c) Use the derivative feature of a graphing utility to confirm your results. f(x) = (x − 2)(x² + 4), (1,-5)
In Exercises(a) Find an equation of the tangent line to the graph of ƒ at the given point (b) Use a graphing utility to graph the function and its tangent line at the point (c) Use the derivative feature of a graphing utility to confirm your results. f(x) = secx, 3' 2
In Exercises(a) Find an equation of the tangent line to the graph of ƒ at the given point (b) Use a graphing utility to graph thefunction and its tangent line at the point (c) Use thederivative feature of a graphing utility to confirm your results. f(x) = (x³ + 4x - 1)(x - 2), (1,-4)
In Exercises find and evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. f(1) 1 4. (x2 – 3x)2 1 16
In Exercises, find the derivative of the function. y sin 3√x + √/sin x =
Find all points on the circle x² + y² = 100 where the slope is 3/4.
In Exercises find and evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. y = 5/3x³ + 4x, (2, 2)
In Exercises evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. Function f(x) = sin x(sin x + cos x) Point (7.1)
In Exercises use a graphing utility to sketch the intersecting graphs of the equations and show that they are orthogonal. [Two graphs are orthogonal if at their point(s) of intersection, their tangent lines are perpendicular to each other.] x + y = 0 x = sin y
In Exercises find and evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. y = √√x² + 8x, (1,3)
In Exercises use a graphing utility to sketch the intersecting graphs of the equations and show that they are orthogonal. [Two graphs are orthogonal if at their point(s) of intersection, their tangent lines are perpendicular to each other.] y² = x³ 2x² + 3y² = 5
In Exercises, find the derivative of the function. y = √√x + sin(2x)² 1/
In Exercises evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. Function h(t) = sec t t Point TT, 1 TT
In Exercises, find the derivative of the function. y = cos sin(tan 77x)
In Exercises evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. Function f(x) = tan x cotx Point (1, 1)
In Exercises verify that the two families of curves are orthogonal, where C and K are real numbers. Use a graphing utility to graph the two families for two values of C and two values of K. x² + y² = C², y = kx
In Exercises verify that the two families of curves are orthogonal, where C and K are real numbers. Use a graphing utility to graph the two families for two values of C and two values of K. xy = C, x² - y² = K
In Exercises use a graphing utility to sketch the intersecting graphs of the equations and show that they are orthogonal. [Two graphs are orthogonal if at their point(s) of intersection, their tangent lines are perpendicular to each other.] 2x² + y² = 6 y² = 4x
In Exercises use a graphing utility to sketch the intersecting graphs of the equations and show that they are orthogonal. [Two graphs are orthogonal if at their point(s) of intersection, their tangent lines are perpendicular to each other.] x³ = 3( y − 1) - x(3y - 29) = 3
In your own words, state the guidelines for implicit differentiation.
Describe the difference between the explicit form of a function and an implicit equation. Give an example of each.
In Exercises, find the derivative of the function.y = sin(tan 2x)
In Exercises find the points at which the graph of the equation has a vertical or horizontal tangent line. 4x² + y² - 8x + 4y + 4 = 0
In Exercises evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. Function y 1 + csc x 1 - csc x Point 6 -3
In Exercises, find the derivative of the function.y = 3x - 5 cos(πX)²
In Exercises, find the derivative of the function.ƒ(t) = 3 sec²(πt - 1)
In Exercises use a computer algebra system to find the derivative of the function. f(0) = sin 0 1 - cos 0
In Exercises, find the derivative of the function. f(0) = sin² 20
In Exercises use a computer algebra system to find the derivative of the function. g(0) 0 1 - sin 0
In Exercises, find the derivative of the function.h(t) = 2 cot²(πt + 2)
In Exercises use a computer algebra system to find the derivative of the function. f(x) x²-x-3 I + zx (x² + x + 1)
In Exercises, find the derivative of the function. g(θ) = cos² 8θ
In Exercises use a computer algebra system to find the derivative of the function. g(x) = x + 1 x + 2, (2x - 5)
Show that the normal line at any point on the circle x² + y2 = r² passes through the origin.
In Exercises find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point.) Use a graphing utility to graph the equation, tangent line, and normal line. x² + y² = 36 (6,0), (5, √11)
In Exercises, find the derivative of the function.ƒ(θ) = tan² 5θ
In Exercises find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point.) Use a graphing utility to graph the equation, tangent line, and normal line. x² + y² = 25 (4,3), (-3, 4)
In Exercises, find the derivative of the function. g(t) = 5 cos² πt
In exercises find the derivative of the trigonometric function. h(0) = 50 sec 0 + 0 tan 0
In Exercises, find the derivative of the function.y = 4 sec² x
In exercises find the derivative of the trigonometric function. y = 2x sin x + x² cos.
In Exercises, find the derivative of the function. g(v) = COS V CSC V
In exercises find the derivative of the trigonometric function. f(x) = sin x cos x
In Exercises use a graphing utility to graph the equation. Find an equation of the tangent line to the graph at the given point and graph the tangent line in the same viewing window. y²: x-1 x² + 1' 2, √√5 5
In Exercises, find the derivative of the function. f(x) = cot x sin x
In exercises find the derivative of the trigonometric function. f(x) = x² tan x
In Exercises use a graphing utility to graph the equation. Find an equation of the tangent line to the graph at the given point and graph the tangent line in the same viewing window. √x + √y = 5, (9,4)
In Exercises, find the derivative of the function. g(0) = sec (0) tan(¹0)
In exercises find the derivative of the trigonometric function. y = x sin x + cos x
In exercises find the derivative of the trigonometric function. y=-cscx sin x -
In Exercises find d²y/dx² implicitly in terms of x and y.y3 = 4x
In Exercises find d²y/dx² implicitly in terms of x and y.y² = x³
In Exercises, find the derivative of the function.h(x) = sin 2x cos 2x
In Exercises find d²y/dx² implicitly in terms of x and y.xy - 1 = 2x + y²
In Exercises, find the derivative of the function.y = cos(1-2x)²
In exercises find the derivative of the trigonometric function. y = sec x X
In Exercises find an equation of the tangent line to the graph at the given point. Then use a graphing utility to graph the function and its tangent line in the same viewing window.Bullet-nose curve + -3-2-1 4 3 2 1 -2+ f(x) = |x| √2-x² (1, 1) |||X 1 2 3
In Exercises(a) Find an equation of the tangent line to the graph off at the given point(b) Use a graphing utility to graph the function and its tangent line at the point f(x) = (9-x²)2/3, (1,4)
In Exercises(a) Find an equation of the tangent line to the graph off at the given point(b) Use a graphing utility to graph the function and its tangent line at the point f(x) = sin 2x, (7,0)
In Exercises determine the point(s) at which the graph of the function has a horizontal tangent line. f(x) = x - 4 x² - 7
In Exercises(a) Find an equation of the tangent line to the graph off at the given point(b) Use a graphing utility to graph the function and its tangent line at the point f(x) = tan² x,
In Exercises(a) Find an equation of the tangent line to the graph off at the given point(b) Use a graphing utility to graph the function and its tangent line at the point y = cos 3x, ㅠ 4' 2) 2
Find equations of the tangent lines to the graph of ƒ(x) = (x + 1)/(x − 1) that are parallel to the line2y + x = 6. Then graph the function and the tangent lines.
In Exercises(a) Find an equation of the tangent line to the graph off at the given point(b) Use a graphing utility to graph the function and its tangent line at the point y = 2 tan³ x, (4.2)
In Exercises verify that ƒ'(x) = g'(x), and explain the relationship between ƒ and g. f(x) 3x x + 2' g(x) 5x+4 x + 2
In Exercises find an equation of the tangent line to the graph at the given point. Then use a graphing utility to graph the function and its tangent line in the same viewing window.Top half of circle + -6-4-2 f(x) = √25 - x2 y 8 6 42 -4 (3,4) 114x 2 4 6
In Exercises verify that ƒ'(x) = g'(x), and explain the relationship between ƒ and g. f(x) = = sin x X 3x g(x): = sin x + 2x X
Find equations of the tangent lines to the graph of ƒ(x) = x/(x - 1) that pass through the point (-1,5).Then graph the function and the tangent lines.
In Exercises use the graphs of ƒ and g. Let p(x) = ƒ(x)g(x) and q(x) = ƒ(x)/g(x).(a) Find P' (1).(b) Find q' (4). 10- --8-- ·2· ### +2+4 6+8+10 -2 X
In Exercises use the graphs of ƒ and g. Let p(x) = ƒ(x)g(x) and q(x) = ƒ(x)/g(x).(a) Find P' (4).(b) Find q' (7). -10-- ## INE ENE -22 4 6 8 10 X
Determine the point(s) at which the graph ofhas a horizontal tangent. f(x)= = X 2x - 1
Determine the point(s) in the interval (0, 2π) at which the graph of ƒ(x) = 2 cos x + sin 2x has a horizontal tangent.
The length of a rectangle is given by 6t + 5 and its height is √t, where t is time in seconds and the dimensions arein centimeters. Find the rate of change of the area with respectto time.
The ordering and transportation cost C for the components used in manufacturing a product iswhere C is measured in thousands of dollars and x is the order size in hundreds. Find the rate of change of C with respect to x when (a) x = 10 (b) x = 15 (c) x = 20 What do these rates
The radius of a right circular cylinder is given by √t + 2 and its height is 1/2√t, where t is time in seconds and the dimensions are in inches. Find the rate of change of the volume with respect to time.
In Exercises find the second derivative of the function.ƒ(x) = 5(2 - 7x)4
In Exercises find the second derivative of the function.ƒ(x) = 6(x³ + 4)³
A population of 500 bacteria is introduced into a culture and grows in number according to the equationwhere t is measured in hours. Find the rate at which the population is growing when t = 2. P(t) = 500(1 + 4t 50+ 1²
Prove the following differentiation rules.(a)(b)(c) d dx -[sec x] = sec x tan x
In Exercises find the second derivative of the function. f(x) 1 9-x
In Exercises find the second derivative of the function. f(x) = 8 (x - 2)²
In Exercises evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. h(x) = 3x + 1), (1, 64 9
In Exercises find the second derivative of the function. f(x) = x² + 2x³ - 3x² - x
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