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mathematics
calculus early transcendentals
Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions
Verify the following derivative formulas using the Quotient Rule.d/dx (sec x) = sec x tan x
Verify the following derivative formulas using the Quotient Rule.d/dx (cot x) = -csc2 x
Find dy/dx for the following functions.y = x sin x / 1 + cos x
Find dy/dx for the following functions.y = cos2 x
Find dy/dx for the following functions.y = (x2 - 1) sin x / sin x + 1
Find dy/dx for the following functions.y = sin x cos x
Find dy/dx for the following functions.y = 1 - sin x/1 + sin x
Find dy/dx for the following functions.y = cos x / sin x + 1
Find dy/dx for the following functions.y = e6x sin x
Find dy/dx for the following functions.y = x sin x
Find dy/dx for the following functions.y = sin x + 4e0.5x
Find dy/dx for the following functions.y = e-x sin x
Find dy/dx for the following functions.y = 5x2 + cos x
Find dy/dx for the following functions.y = sin x + cos x
Use Theorem 3.11 to evaluate the following limits. sin (x + 3) lim x→-3 x + 8x + 15 .2 х- 8х + 15
Use Theorem 3.11 to evaluate the following limits. sin (x lim x→2 x - 4 2) .2
Use Theorem 3.11 to evaluate the following limits. sec 0 – 1 lim ө
Use Theorem 3.11 to evaluate the following limits. tan 7x lim x→0 sin x
Use Theorem 3.11 to evaluate the following limits. - 1 sec 0 lim 0→0 Ө
Use Theorem 3.11 to evaluate the following limits. tan 5x lim х-—0
Use Theorem 3.11 to evaluate the following limits. sin 3x lim x→0 tan 4x
Use Theorem 3.11 to evaluate the following limits. sin 7x lim x0 sin 3x X-
Use Theorem 3.11 to evaluate the following limits. sin 5x lim x0 3x
Use Theorem 3.11 to evaluate the following limits. sin 3x lim х х
Where does the graph of sin x have a horizontal tangent line? Where does cos x have a value of zero? Explain the connection between these two observations.
Let f(x) = sin x. What is the value of f'(π)?
How can you use the derivatives d/dx (sin x) = cos x, d/dx (tan x) = sec2 x, and d/dx (sec x) = sec x tan x to remember the derivatives of cos x, cot x, and csc x?
Explain why the Quotient Rule is used to determine the derivative of tan x and cot x.
Why is it not possible to evaluateby direct substitution? sin x lim
One of several Leibniz Rules in calculus deals with higher-order derivatives of products. Let (fg)(n) denote the nth derivative of the product fg, for n ≥ 1.a. Prove that (fg)(2) = f"g + 2 f'g' + fg".b. Prove that, in general,
Assume that f, g, and h are differentiable at x.a. Use the Product Rule (twice) to find a formula for d/dx (f(x)g(x)h(x)).b. Use the formula in (a) to find d/dx (e2x (x - 1)(x + 3)).
Assuming the first and second derivatives of f and g exist at x, find a formula for d? (f(x) dx² \g(x). dr2
Use the Quotient Rule and Exercise 88 to show that d/dx (ekx) = kekx, for negative integers k.Data from exercise 88Proof by induction is a method in which one begins by showing that a statement, which involves positive integers, is true for a particular value (usually k = 1). In the second step,
Proof by induction is a method in which one begins by showing that a statement, which involves positive integers, is true for a particular value (usually k = 1). In the second step, the statement is assumed to be true for k = n, and the statement is proved for k = n + 1, which concludes the
Assuming the first and second derivatives of f and g exist at x, find a formula for d2/dx2 (f(x)g(x)).
Let F = f/g be the quotient of two functions that are differentiable at x.a. Use the definition of F' to show thatb. Now add -f(x)g(x) + f(x)g(x) (which equals 0) to the numerator in the preceding limit to obtainUse this limit to obtain the Quotient Rule.c. Explain why F' = (f/g)' exists, whenever
Suppose f is differentiable on an interval containing a and b, and let P(a, f(a)) and Q(b, f(b)) be distinct points on the graph of f. Let c be the x-coordinate of the point at which the lines tangent to the curve at P and Q intersect, assuming that the tangent lines are not parallel (see
In general, the derivative of a quotient is not the quotient of the derivatives. Find non constant functions f and g such that the derivative of f/g equals f'/g'.
In general, the derivative of a product is not the product of the derivatives. Find non constant functions f and g such that the derivative of fg equals f'g'.
The magnitude of the gravitational force between two objects of mass M and m is given by F(x) = -GMm/x2, where x is the distance between the centers of mass of the objects and G = 6.7 × 10-11 N-m2/kg2 is the gravitational constant (N stands for newton, the unit of force; the negative sign
The magnitude of the electrostatic force between two point charges Q and q of the same sign is given by F(x) = kQq/x2, where x is the distance (measured in meters) between the charges and k = 9 × 109 N-m2/C2 is a physical constant (C stands for coulomb, the unit of charge; N stands for newton, the
Suppose the line tangent to the graph of f at x = 2 is y = 4x + 1 and suppose y = 3x - 2 is the line tangent to the graph of g at x = 2. Find an equation of the line tangent to the following curves at x = 2.a. y = f(x)g(x)b. y = f(x)/g(x)
Use the following table to find the given derivatives.
Use the following table to find the given derivatives.
Use the following table to find the given derivatives.
Use the following table to find the given derivatives.
Use the following table to find the given derivatives.
Use the following table to find the given derivatives.
f(2) = 2 and f'(2) = 3. Let g(x) = x2 • f(x) and h(x) = f(x)/x - 3.a. Find an equation of the line tangent to y = g(x) at x = 2.b. Find an equation of the line tangent to y = h(x) at x = 2.
Use any method to evaluate the derivative of the following functions.h(x) = (5x7 + 5x)(6x3 + 3x2 + 3)
Use any method to evaluate the derivative of the following functions.where a is a positive constant. х — а Vx - Va
Use any method to evaluate the derivative of the following functions. 2 - r - Vr r + 1 h(r)
Use any method to evaluate the derivative of the following functions. 2z f(z) = z(e% + 4) - z* + 1 ,3z .2
Use any method to evaluate the derivative of the following functions. 2х f(x) = 4x? 5х + 1
Use any method to evaluate the derivative of the following functions. .2 4 — х f(x) х — 2
Find f'(x) and f"(x).f(x) = x2 - 7x/x + 1
Find f'(x) and f"(x).f(x) = x/x + 2
Find f'(x), f"(x), and f"'(x).f(x) = x2e3x
Find f'(x), f"(x), and f"'(x).f(x) = 1/x
Determine whether the following statements are true and give an explanation or counterexample.a. d/dx (e5) = 5e4.b. The Quotient Rule must be used to evaluate d/dx (x2 + 3x + 2/x).c. d/dx (1/x5) = 1/5x4.d. dn/dxn (e3x) = 3ne3x, for any integer n ≥ 1.
Compute the derivative of the following functions. х+ 1 h(x) .2 x?ет
Compute the derivative of the following functions. xe* h(x) X + 1
Compute the derivative of the following functions. (x – 1)(2x² – 1) h(x) 1) х3 — 1
Compute the derivative of the following functions. (x + 1)e* 8(x) х — 2
Let f(t) = 100e-0.05t.a. Find the values of t for which the slope of the curve y = f(t) is -5.b. Does the graph of f have a horizontal tangent line?
Let f(x) = xe2x.a. Find the values of x for which the slope of the curve y = f(x) is 0.b. Explain the meaning of your answer to part (a) in terms of the graph of f.
A $200 investment in a savings account grows according to A(t) = 200e0.0398t, for t ≥ 0, where t is measured in years.a. Find the balance of the account after 10 years.b. How fast is the account growing (in dollars>year) at t = 10?c. Use your answers to parts (a) and (b) to write the equation
The half-life of an antibiotic in the bloodstream is 10 hours. If an initial dose of 20 milligrams is administered, the quantity left after t hours is modeled by Q(t) = 20e-0.0693t, for t ≥ 0.a. Find the instantaneous rate of change of the amount of antibiotic in the bloodstream, for t ≥ 0.b.
Consider the following population functions.a. Find the instantaneous growth rate of the population, for t ≥ 0.b. What is the instantaneous growth rate at t = 5?c. Estimate the time when the instantaneous growth rate is the greatest.d. Evaluate and interprete. Use a graphing utility to graph the
Consider the following population functions.a. Find the instantaneous growth rate of the population, for t ≥ 0.b. What is the instantaneous growth rate at t = 5?c. Estimate the time when the instantaneous growth rate is the greatest.d. Evaluate and interprete. Use a graphing utility to graph the
Compute the derivative of the following functions.A = 2500e0.075t
Compute the derivative of the following functions.y = 2ex + 3e-x/3
Compute the derivative of the following functions.f(x) = (1 - 2x)2e-x
Compute the derivative of the following functions.g(x) = x/e3x
Compute the derivative of the following functions.y = 3x2 - 2x + e-2x
Compute the derivative of the following functions.f(x) = 15e3x
Compute the derivative of the following functions.g(t) = 2tet/2
Compute the derivative of the following functions.f(x) = xe7x
Find the derivative of the following functions. 4х3 + 3х + 1 p(x)
Find the derivative of the following functions. t3 + 3t² + t 8(1) 13 ||
Find the derivative of the following functions. w4 + 5w2 + w
Find the derivative of the following functions.g(t) = 3t2 + 6/t7
Find the derivative of the following functions.y = 4/p3
Find the derivative of the following functions.f(x) = 3x-9
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/x; a = 1
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 = 1 + 2x + xex; 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. 2r2 1' a = 1| У — Зх —
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. х+ 5 -; a = 3 y - 1 х —
a. Use the Quotient Rule to find the derivative of the given function. Simplify your result.b. Find the derivative by first simplifying the function. Verify that your answer agrees with part (a).where a is a constant. 2ах + a* x? У .2 х — а
a. Use the Quotient Rule to find the derivative of the given function. Simplify your result.b. Find the derivative by first simplifying the function. Verify that your answer agrees with part (a).where a is a constant. x² – a? х х — а
a. Use the Quotient Rule to find the derivative of the given function. Simplify your result.b. Find the derivative by first simplifying the function. Verify that your answer agrees with part (a). 4s3 – 8s2 + 4s 8.s 4.s 4s
a. Use the Quotient Rule to find the derivative of the given function. Simplify your result.b. Find the derivative by first simplifying the function. Verify that your answer agrees with part (a). f(w) и»3
Find the derivative of the following functions.y = (2√x - 1)(4x + 1)-1
Find the derivative of the following functions.g(x) = ex/x2 - 1
Find the derivative of the following functions. - 1 h(w) w? + 1
Find the derivative of the following functions.y = (3t - 1)(2t - 2)-1
Find the derivative of the following functions.f(x) = e-x√x
Find the derivative of the following functions. 2e* – 1 f(x) 2e* + 1
Find the derivative of the following functions.f(x) = xe-x
Find the derivative of the following functions. et f(x) e* + 1
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