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mathematics
calculus early transcendentals

Questions and Answers of Calculus Early Transcendentals

Find the derivative of the following functions. x³ – 4x² + x f(x)
Find the derivative of the following functions. f(x) x + 1
a. Use the Product Rule to find the derivative of the given function. Simplify your result.b. Find the derivative by expanding the product first. Verify that your answer agrees with part (a).h(z) =
a. Use the Product Rule to find the derivative of the given function. Simplify your result.b. Find the derivative by expanding the product first. Verify that your answer agrees with part (a).g(y) =
a. Use the Product Rule to find the derivative of the given function. Simplify your result.b. Find the derivative by expanding the product first. Verify that your answer agrees with part (a).y = (t2
a. Use the Product Rule to find the derivative of the given function. Simplify your result.b. Find the derivative by expanding the product first. Verify that your answer agrees with part (a).f(x) =
Find the derivative of the following functions.s(t) = 4et√t
Find the derivative of the following functions.g(w) = ew(w3 - 1)
Find the derivative of the following functions.f(x) = (1 + 1/x2)(x2 + 1)
Find the derivative of the following functions.h(x) = (x - 1)(x3 + x2 + x + 1)
Find the derivative of the following functions.g(w) = ew(5w2 + 3w + 1)
Find the derivative of the following functions.f(t) = t5et
Find the derivative of the following functions.g(x) = 6x - 2xex
Find the derivative of the following functions.f(x) = 3x4 (2x2 - 1)
Show two ways to differentiate f(x) = (x - 3)(x2 + 4).
What is the derivative of y = ekx? For what values of k does this rule apply?
Show two ways to differentiate f(x) = 1/x10.
State the Extended Power Rule for differentiating xn. For what values of n does the rule apply?
How do you find the derivative of the quotient of two functions that are differentiable at a point?
How do you find the derivative of the product of two functions that are differentiable at a point? How do you find the derivative of the quotient of two functions that are differentiable at a point?
Computing the derivative of f(x) = x2exa. Use the definition of the derivative to show thatb. Manipulate the limit in part (a) to arrive at f'(x) = ex(x2 + 2x). Use the fact that h?)eh – + 2xh +
Computing the derivative of f(x) = e2xa. Use the definition of the derivative to show that
Computing the derivative of f(x) = e-xa. Use the definition of the derivative to show thatb. Show that the limit in part (a) is equal to –1. Use the facts that and ex is continuous for all x.c. Use
a. Explain why the Power Rule is consistent with the formulab. Prove that the Power Rule holds for n = 3/2. Use the definition of the derivative: d/dx (x3/2) = c. Prove that the Power Rule holds for
Suppose n is a negative integer and f(x) = xn. Use the following steps to prove that f'(a) = nan - 1, which means the Power Rule for positive integers extends to all integers.a. Assume that m = - n,
The Binomial Theorem states that for any positive integer n,Use this formula and the definitionto show that d/dx (xn) = nxn - 1, for any positive integer n. п(n — 1) (a + b)" = a" + na"-'b + 2•1
For the constant function f(x) = c, use the definition of the derivative to show that f'(x) = 0.
City planners model the size of their city using the function A(t) = - 1/50t2 + 2t + 20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.a. Compute
When observations begin at t = 0, a cell culture has 1200 cells and continues to grow according to the function p(t) = 1200 et, where p is the number of cells and t is measured in days.a. Compute
The distance an object falls (when released from rest, under the influence of Earth’s gravity, and with no air resistance) is given by d(t) = 16t2, where d is measured in feet and t is measured in
The position of a small rocket that is launched vertically upward is given by s(t) = -5t2 + 40t + 100, for 0 ≤ t ≤ 10, where t is measured in seconds and s is measured in meters above the
The limitis the derivative of a function f at a point a. Find one possible f and a, and evaluate the limit. et lim
Use a calculator to approximate the following limits. х lim x→0* \x
Use a calculator to approximate the following limits. lim x* x→0+ -X
Use a calculator to approximate the following limits. п lim п п- 00
Use a calculator to approximate the following limits. e3x – 1 lim х х—0
Complete the following table and give approximations for 34 – 1 - 1 and lim - lim 24 3h - -1.0 -0.1 -0.01 -0.001 -0.0001 -0.00001
The following limits represent f'(a) for some function f and some real number a.a. Find a possible function f and number a.b. Evaluate the limit by computing f'(a). 100 х lim х
The following limits represent f'(a) for some function f and some real number a.a. Find a possible function f and number a.b. Evaluate the limit by computing f'(a). (1 + h)š + (1 + h)³ lim
The following limits represent f'(a) for some function f and some real number a.a. Find a possible function f and number a.b. Evaluate the limit by computing f'(a). V9 + h – Vỹ lim
Use the table to find the following derivatives. 3 4 х 5 3 f'(x) 3 5 g'(x) (2x – 3g(x)) dx
Use the table to find the following derivatives. 3 4 х 5 3 f'(x) 3 5 g'(x) (1.5 f(x)) dx sл) x=2
Use the table to find the following derivatives. 3 4 х 5 3 f'(x) 3 5 g'(x) (f(x) + g(x)) dx |x=1
Let F = f + g and G = 3f - g, where the graphs of f and g are shown in the figure. Find the following derivatives.G'(5) УА y, y = f(x) 3 y = g(x) %3D 1+ х 5 3.
Let F = f + g and G = 3f - g, where the graphs of f and g are shown in the figure. Find the following derivatives.F'(5) УА y, y = f(x) 3 y = g(x) %3D 1+ х 5 3.
Let F = f + g and G = 3f - g, where the graphs of f and g are shown in the figure. Find the following derivatives.G'(2) УА y, y = f(x) 3 y = g(x) %3D 1+ х 5 3.
Let F = f + g and G = 3f - g, where the graphs of f and g are shown in the figure. Find the following derivatives.F'(2) УА y, y = f(x) 3 y = g(x) %3D 1+ х 5 3.
Determine the constants b and c such that the line tangent to f(x) = x2 + bx + c at x = 1 is y = 4x + 2.
Find the equation of the line tangent to the curve y = x + √x that has slope 2.
Suppose the line tangent to the graph of f at x = 2 is y = 4x + 1 and suppose the line tangent to the graph of g at x = 2 has slope 3 and passes through (0, -2). Find an equation of the line tangent
Suppose f(3) = 1 and f'(3) = 4. Let g(x) = x2 + f(x) and h(x) = 3f(x).a. Find an equation of the line tangent to y = g(x) at x = 3.b. Find an equation of the line tangent to y = h(x) at x = 3.
Determine whether the following statements are true and give an explanation or counterexample.a. d/dx(10)5 = 5 •104.b. The slope of a line tangent to f(x) = ex is never 0.c. d/dx (e3) = e3.d. d/dx
Find f'(x), f"(x), and f"'(x) for the following functions.f(x) = 10ex
Find f'(x), f"(x), and f"'(x) for the following functions. х? — 7х — 8 f(x) x + 1
Find f'(x), f"(x), and f"'(x) for the following functions.f(x) = 3x2 + 5ex
Find f'(x), f"(x), and f"'(x) for the following functions.f(x) = 5x4 + 10x3 + 3x + 6
Find f'(x), f"(x), and f"'(x) for the following functions.f(x) = 3x3 + 5x2 + 6x
Let f(x) = 4√x - x.a. Find all points on the graph of f at which the tangent line is horizontal.b. Find all points on the graph of f at which the tangent line has slope -1/2.
Let f(x) = 2ex - 6x.a. Find all points on the graph of f at which the tangent line is horizontal.b. Find all points on the graph of f at which the tangent line has slope 12.
Let f(x) = 2x3 - 3x2 - 12x + 4.a. Find all points on the graph of f at which the tangent line is horizontal.b. Find all points on the graph of f at which the tangent line has slope 60.
Let f(t) = t3 - 27t + 5.a. Find the values of t for which the slope of the curve y = f(t) is 0.b. Find the values of t for which the slope of the curve y = f(t) is 21.
Let f(x) = x2 - 6x + 5.a. Find the values of x for which the slope of the curve y = f(x) is 0.b. Find the values of x for which the slope of the curve y = f(x) is 2.
a. Find an equation of the line tangent to the given curve at a.b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.y = ex/4 - x; a = 0
a. Find an equation of the line tangent to the given curve at a.b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.y = ex; a = ln 3
a. Find an equation of the line tangent to the given curve at a.b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.y = x3 - 4x2 + 2x - 1; a = 2
a. Find an equation of the line tangent to the given curve at a.b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.y = -3x2 + 2; a = 1
Find the derivative of the following functions by first simplifying the expression.a is a constant. 2ах + a? у з х — а ||
Find the derivative of the following functions by first simplifying the expression.a is a positive constant. х — а Vx - Va
Find the derivative of the following functions by first simplifying the expression. х3 — бх? + 8х h(x) .2 2х
Find the derivative of the following functions by first simplifying the expression. х2 — 1 8(х) х — 1
Find the derivative of the following functions by first simplifying the expression. 12s3 У - ,3 8s2 + 12s 4s
Find the derivative of the following functions by first simplifying the expression. w3 f(w)
Find the derivative of the following functions by first expanding the expression. Simplify your answers.h(x) = √x (√x - 1)
Find the derivative of the following functions by first expanding the expression. Simplify your answers.h(x) = (x2 + 1)2
Find the derivative of the following functions by first expanding the expression. Simplify your answers.g(r) = (5r3 + 3r + 1)(r2 + 3)
Find the derivative of the following functions by first expanding the expression. Simplify your answers.f(x) = (2x + 1)(3x2 + 2)
Find the derivative of the following functions.s(t) = 4√t - 1/4t4 + t + 1
Find the derivative of the following functions.g(w) = 2w3 + 3w2 + 10w
Find the derivative of the following functions.f(t) = 6√t - 4t3 + 9
Find the derivative of the following functions.f(x) = 10x4 - 32x + e2
Find the derivative of the following functions.g(x) = 6x5 - x
Find the derivative of the following functions.f(x) = 3x4 + 7x
Find the derivative of the following functions.f(s) = √s/4
Find the derivative of the following functions.g(t) = 100t2
Find the derivative of the following functions.g(t) = 6√t
Find the derivative of the following functions.p(x) = 8x
Find the derivative of the following functions.g(w) = 5/6 w12
Find the derivative of the following functions.f(x) = 5x3
Find the derivative of the following functions.f(v) = v100
Find the derivative of the following functions.h(t) = t
Find the derivative of the following functions.g(x) = e3
Find the derivative of the following functions.f(x) = 5
Find the derivative of the following functions.f(t) = t11
Find the derivative of the following functions.y = x5.
Assume the derivatives of f and g exist in the following exercise.How do you find the fifth derivative of a function?
Assume the derivatives of f and g exist in the following exercise.How do you find the derivative of a constant multiplied by a function?
Assume the derivatives of f and g exist in the following exercise.How do you find the derivative of the sum of two functions f + g?
Assume the derivatives of f and g exist in the following exercise.Give a nonzero function that is its own derivative.
Assume the derivatives of f and g exist in the following exercise.In this section, we showed that the rule d/dx (xn) = nxn - 1 is valid for what values of n?
Assume the derivatives of f and g exist in the following exercise.If the limit definition of a derivative can be used to find f', then what is the purpose of using other rules to find f'?

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