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study help
mathematics
precalculus
Questions and Answers of
Precalculus
A balloon rises at a rate of 4 meters per second from a point on the ground 50 meters from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the
The table shows the numbers (in millions) of participants in the free lunch program f and the reduced price lunch program r in the United States for the years 2007 through 2014. (a) Use the
Find the second derivative of the function.h(x) = 6x−2 + 7x2
Find the derivative of the trigonometric function.f(θ) = 1/4 sin2 2θ
Find the second derivative of the function.f(x) = 15x5/2
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) = tan x cotx Point (1, 1)
Find the second derivative of the function.f(x) = 20 5√x
The figure shows the graph of g′.(a)(b)(c) What can you conclude about the graph of g knowing that g′(1) = −8/3?(d) What can you conclude about the graph of g knowing that g′(−4) = 7/3?(e)
Find the second derivative of the function.f(θ) = 3 tan θ
Find the second derivative of the function.h(t) = −12 csc t
Find equations of the two tangent lines to the graph of f that pass through the indicated point. f(x) = x² 10 064 + -6-4-2 8 -4 2 4 6 (1, -3)
Find the derivative of the function.y = (x2 − 6)3
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) = (sin x) (sin x + cos x) Point (41)
Find the derivative of the function.y = 1/(x2 + 5)3
Identify a function f that has the given characteristics. Then sketch the function.f(0) = 4; f′(0) = 0; f′(x) < 0 for x < 0; f′(x) > 0 for x > 0
Find the derivative of the function.y = −6 sin 3x4
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.y = 5√3x3 + 4x, (2, 2)
(a) Find an equation of the tangent line to the graph of f at the given point,(b) Use a graphing utility to graph the function and its tangent line at the point, and(c) Use the tangent feature of a
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 < 2
Find the derivative of the function.f(s) = (s2 − 1)5/2 (s3 + 5)
Find the derivative of the function.f(x) = (x/√x + 5)3
Consider the function f(x) = 1/3x3.(a) Use a graphing utility to graph the function and estimate the values of f′(0), f′(1/2), f′(1), f′(2), and f′(3).(b) Use your results from part (a) to
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.f(x) = 1/ (x2 − 3x)2, (4, 1/16)
(a) Find an equation of the tangent line to the graph of f at the given point,(b) Use a graphing utility to graph the function and its tangent line at the point, and(c) Use the tangent feature of a
Use the graph to answer the questions.(a) Which is greater, the slope of the tangent line at x = −3 or the slope of the tangent line at x = −1?(b) Estimate the point(s) where the graph has a
Find the derivative of the function.h(x) = (x + 5/x2 + 3)2
Find the slope of the graph of the function at the given point.f(x) = √1 − x3, (−2, 3)
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.y = 4 / (x + 2)2, (0, 1)
Find the slope of the graph of the function at the given point.f(x) = 3√x2 − 1, (3, 2)
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.y = 4 / (x2 − 2x)3, (1, −4)
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. y X + COS X. l'a 2 al
Find the slope of the graph of the function at the given point.f(x) = x + 8 /√3x + 1, (0, 8)
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.y = 26 − sec3 4x, (0, 25)
The graph shows the normal lines from the point (2, 0) to the graph of the parabola x = y2. How many normal lines are there from the point (x0, 0) to the graph of the parabola if(a) x0 = 1/4,(b) x0 =
Find the slope of the graph of the function at the given point.f(x) = 3x + 1/(4x − 3)3, (1, 4)
Describe the x-values at which f is differentiable. f(x) -4 = r x2 – 4 - 5432 y 34 X
The relationship between f and g is given. Explain the relationship between f′ and g′.g(x) = 2 f(x)
(a) Find an equation of the tangent line to the graph of the function at the given point,(b) Use a graphing utility to graph the function and its tangent line at the point, and(c) Use the tangent
(a) Find an equation of the tangent line to the graph of the function at the given point,(b) Use a graphing utility to graph the function and its tangent line at the point, and(c) Use the tangent
The relationship between f and g is given. Explain the relationship between f′ and g′.g(x) = 3 f(x) − 1
Find all points on the circle x2 + y2 = 100 where the slope is 3/4.
(a) Find an equation of the tangent line to the graph of the function at the given point,(b) Use a graphing utility to graph the function and its tangent line at the point, and(c) Use the tangent
(a) Find an equation of the tangent line to the graph of the function at the given point,(b) Use a graphing utility to graph the function and its tangent line at the point, and(c) Use the tangent
Find the second derivative of the function.y = x sin2 x
Determine the point(s) at which the graph ofhas a horizontal tangent. f(x) = - 4x 2x - 1
Find dy/dx by implicit differentiation.x2 + y2 = 64
Find dy/dx by implicit differentiation.x3 y − xy3 = 4
Find dy/dx by implicit differentiation.√xy = x − 4y
Find the second derivative of the function. f(x) = 8 (x - 2)²
Find dy/dx by implicit differentiation.x sin y = y cos x
Find dy/dx by implicit differentiation.cos(x + y) = x
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, [%]
Find the second derivative of the function.f(x) = 6(x3 + 4)3
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) = then f'(x) t 1 nx"-1"
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 = xa+2 + bx, then dy/dx = (a + 2)xa+1 + b.
When satellites observe Earth, they can scan only part of Earth’s surface. Some satellites have sensors that can measure the angle shown in the figure. Let h represent the satellite’s distance
Evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. g(t) = tan 2t, 6' √√3)
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) = −g(x) + b, then f′(x) = −g′(x).
Consider the function(a) In general, how do you find the derivative of h(x) = f(x)/g(x) using the Product Rule, where g is a composite function?(b) Find r′(x) using the Product Rule.(c) Find
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 − x)2/3
Find the second derivative of the function.f(x) = sec2 x
Consider the functions f(x) = x2 and g(x) = x3.(a) Graph f and f′ on the same set of axes.(b) Graph g and g′ on the same set of axes.(c) Identify a pattern between f and g and their respective
The cost C (in dollars) of producing x units of a product is C = 60x + 1350. For one week, management determined that the number of units produced x at the end of t hours can be modeled by x =
Find the second derivative of the function.f(x) = x2 + 3x−3
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If a function is continuous at a point, then it is differentiable at that point.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If a function is differentiable at a point, then it is continuous at that point.
Use the position function s(t) = −4.9t2 + v0t + s0 for free-falling objects. A rock is dropped from the edge of a cliff that is 214 meters above water.(a) Determine the position and velocity
Find the second derivative of the function.f(x) = x cos x
The graph of f is shown. Sketch the graphs of f′ and f″. To print an enlarged copy of the graph, go to MathGraphs.com. 432 1 -4+ EIN 3Ï X
Find the given higher-order derivative.f (3) (x) = 5√x4 , f (4) (x)
Find the given higher-order derivative.f (4) (t) = t cos t, f (5) (t)
Describe how you would differentiate a piecewise function. Use your approach to find the first and second derivatives of f(x) = x∣x∣. Explain why f″(0) does not exist.
The linear and quadratic approximations of a function f at x = a are P1(x) = f′(a)(x − a) + f (a) and P2(x) = 1/2 f ″(a)(x − a)2 + f′(a)(x − a) + f (a).(a) Find the specified linear and
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. d'y If y = (x + 1)(x + 2)(x+3)(x + 4), then dx³ = 0.
Prove that d/dx [cos x] = −sin x.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The slope of the function f(x) = cos bx at the origin is −b.
Develop a general rule for the nth derivative of xf(x), where f is a differentiable function of x.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If the position function of an object is linear, then its acceleration is zero.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The function f(x) = sin x + c satisfies f(n) = f(n+4) for all integers n ≥ 1.
Which of the functions graphed are one-to-one, and which are not? 0 y y=-3x³ X
Which of the functions graphed are one-to-one, and which are not? - y y = x - x² X
Which of the functions graphed are one-to-one, and which are not? y y = 2|x| X
Are there two functions ƒ and g such that ƒ ∘ g = g ∘ ƒ? Give reasons for your answer.
What is a function? What is its domain? Its range? What is an arrow diagram for a function? Give examples.
Express the radius of a sphere as a function of the sphere’s surface area. Then express the surface area as a function of the volume.
Sketch the given curves together in the appropriate coordinate plane and label each curve with its equation.y = 3x, y = 8x, y = 2-x, y = (1/4)x
Use graphing software to determine which of the given viewing windows displays the most appropriate graph of the specified function.ƒ(x) = x3 - 4x2 - 4x + 16a. [-1, 1] by [-5, 5] b. [-3, 3] by
Are there two functions ƒ and g with the following property? The graphs of ƒ and g are not straight lines but the graph of ƒ ∘ g is a straight line. Give reasons for your answer.
Which of the functions graphed are one-to-one, and which are not? y = int x
What is the graph of a real-valued function of a real variable? What is the vertical line test?
A central angle in a circle of radius 8 is subtended by an arc of length 10π. Find the angle’s radian and degree measures.
If ƒ(x) is odd, can anything be said of g(x) = ƒ(x) - 2? What if ƒ is even instead? Give reasons for your answer.
Which of the functions graphed are one-to-one, and which are not? y 0 y || 18
What is a piecewise-defined function? Give examples.
A hot-air balloon rising straight up from a level field is tracked by a range finder located 500 ft from the point of liftoff. Express the balloon’s height as a function of the angle the line from
Sketch the given curves together in the appropriate coordinate plane and label each curve with its equation.y = 3-t and y = -3t
Use graphing software to determine which of the given viewing windows displays the most appropriate graph of the specified function.ƒ(x) = 25 + 4x - x2a. [-2, 2] by [-2, 2]b. [-2, 6] by [-1, 4]c.
If g(x) is an odd function defined for all values of x, can anything be said about g(0)? Give reasons for your answer.
Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more
What are the important types of functions frequently encountered in calculus? Give an example of each type.
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