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mathematics
precalculus

Questions and Answers of Precalculus

Find the absolute maximum and absolute minimum values of f on the given interval.f (x) = x-2 ln x , [1/2, 4]
Find the absolute maximum and absolute minimum values of f on the given interval.f (t) = √t/1 + t2, [0, 2]
Find the absolute maximum and absolute minimum values of f on the given interval.f (t) = t - 3√t, [-1, 4]
Find the absolute maximum and absolute minimum values of f on the given interval.f (x) = x/x2 - x + 1, [0, 3]
Find the absolute maximum and absolute minimum values of f on the given interval.f (x) = x + 1/x , [0.2, 4]
Find the absolute maximum and absolute minimum values of f on the given interval.f (t) = (t2 - 4)3, [-2, 3]
Find the absolute maximum and absolute minimum values of f on the given interval.f (x) = 3x4 - 4x3 - 12x2 + 1, [-2, 3]
Find the absolute maximum and absolute minimum values of f on the given interval.f (x) = 2x3 - 3x2 - 12x + 1, [-2, 3]
Find the absolute maximum and absolute minimum values of f on the given interval.f (x) = 5 + 54x - 2x3, [0, 4]
Find the absolute maximum and absolute minimum values of f on the given interval.f (x) = 12 + 4x - x2, [0, 5]
A formula for the derivative of a function f is given. How many critical numbers does f have?f' (x) = 100 cos2x/10 + x2 - 1
Find the critical numbers of the function.f (x) = x-2 ln x
Find the critical numbers of the function.f (x) = x2 e-3x
Find the critical numbers of the function.h(t) = 3t - arcsin t
Find the critical numbers of the function.F(x) = x4/5(x - 4)2
Find the critical numbers of the function.t(x) = 3√4 - x2
Find the critical numbers of the function.h(t) = t3/4 - 2t1/4
Find the critical numbers of the function.h(p) = p - 1/p2 + 4
Find the critical numbers of the function.g(y) = y - 1/y2 - y + 1
Find the critical numbers of the function.f (x) = 2x3 + x2 + 2x
Find the critical numbers of the function.f (x) = 2x3 - 3x2 - 36x
Use Newton’s method to find the coordinates, correct to six decimal places, of the point on the parabola y = (x -1)2 that is closest to the origin.
(a) Apply Newton’s method to the equation x2 - a = 0 to derive the following square-root algorithm (used by the ancient Babylonians to compute sa):(b) Use part (a) to compute s1000 correct to six
Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. Зх In(x? + 2) ух? + 1
Use Newton’s method to find all solutions of the equation correct to six decimal places.ln x = 1/x - 3
Use Newton’s method to find all solutions of the equation correct to six decimal places. 2X = 2 - x2
(a) Show that if the profit P(x) is a maximum, then the marginal revenue equals the marginal cost.(b) If C(x) = 16,000 + 500x - 1.6x2 + 0.004x3 is the cost function and p(x) = 1700 - 7x is the demand
What is the smallest possible area of the triangle that is cut off by the first quadrant and whose hypotenuse is tangent to the parabola y = 4 - x2 at some point?
What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve y = 3/x at some point?
An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east
If a resistor of R ohms is connected across a battery of E volts with internal resistance r ohms, then the power (in watts) in the external resistor is If E and r are fixed but R varies, what is
Answer Exercise 37 if one piece is bent into a square and the other into a circle.A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an
A poster is to have an area of 180 in2 with 1-inch margins at the bottom and sides and a 2-inch margin at the top. What dimensions will give the largest printed area?
A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See Exercise 1.1.62.) If the perimeter of the
If the two equal sides of an isosceles triangle have length a, find the length of the third side that maximizes the area of the triangle.
Find f. sec t (sec t + tan t), -7/2 < t
Find f. f'(x) = (x + 1)/x, ƒ(1) = 5
Find the most general antiderivative of the function. |f(x) = 1 + 2 sin x + 3//x
Find the most general antiderivative of the function. r(0) = sec e tan 0 – 2e
Find the most general antiderivative of the function. 1 +t+ t? g(t) =
Find the most general antiderivative of the function. 3t* – t3 + 6t? | f(t) –
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. = | ` In(1 + t²) dt
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. х Vt + t³ dt 9(х)
Find ∫50 f (x) dx if | 3 for x < 3 f(x) = for x > 3
Express the limit as a definite integral. и lim -2 Σ n-0 n = 1 + (i/n)?
Express the limit as a definite integral. lim 2 n- n° i=1
Let f (x) = 0 if x is any rational number and f (x) = 1 if x is any irrational number. Show that f is not integrable on [0, 1].
(a) If f is continuous on [a, b], show that(b) Use the result of part (a) to show that
Use properties of integrals, together with Exercises 27 and 28, to prove the inequality. *T/2 x sin x dx < т 0,
Use properties of integrals, together with Exercises 27 and 28, to prove the inequality. '3 26 > 3 V xª + 1 dx
Use Property 8 to estimate the value of the integral.
Use Property 8 to estimate the value of the integral. хе^dx
Use Property 8 to estimate the value of the integral. (x3 — Зх + 3) dх
Use Property 8 to estimate the value of the integral. Сп/3 tan x dx 7/4
Use Property 8 to estimate the value of the integral. *3 dx Jo x + 4
Use the properties of integrals to verify the inequality without evaluating the integrals. У3 п T/3 т sin x dx T/6 < 12 12 VI
Use the properties of integrals to verify the inequality without evaluating the integrals. 2 < , VI + x² dx < 2/2
Use the properties of integrals to verify the inequality without evaluating the integrals. I V1 + x dx /1 + x² dx <
Use the properties of integrals to verify the inequality without evaluating the integrals. (x? — 4х + 4) dx > 0
Suppose f has absolute minimum value m and absolute maximum value M. Between what two values must lie?Which property of integrals allows you to make your conclusion? Sof(x) dx lie?
If where f is the function whose graph is given, which of the following values is largest? F(x) = f; f(t) dt, (A) F(0) (D) F(3) (B) F(1) (E) F(4) (C) F(2)
Use the result of Exercise 27 and the fact that(from Exercise 5.1.31), together with the properties of integrals, to evaluate *T/2 cos x dx = 1 72 п/2 Г7 (2 сos x — 5х) dx. 0,
Use the properties of integrals and the result of Example 3 to evaluate S; (2e* – 1) dx.
In Example 5.1.2 we showed that Use this fact and the properties of integrals to evaluate So x² dx
Given that
Evaluate | V1 + x* dx.
Evaluate the integral by interpreting it in terms of areas. C12x – 1|dx
Evaluate the integral by interpreting it in terms of areas. xp |x| -4
Evaluate the integral by interpreting it in terms of areas. 25 — х?) dx (х -5
Evaluate the integral by interpreting it in terms of areas. Гx - 2) dx
Evaluate the integral by interpreting it in terms of areas. L,а — х) dх
The graph of gconsists of two straight lines and a semi-circle. Use it to evaluate each integral. | g(x) dx (b) (c) g(x) dx | 9(x) dx |(a)
Express the integral as a limit of Riemann sums. Do not evaluate the limit. г( C5 dx 12 х
Express the integral as a limit of Riemann sums. Do not evaluate the limit. 3 /4 + x² dx
Prove that 3 b3 – a3 x² dx 3
Prove that b? – a? х dx
Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. — 3х?) dx
Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. (2х —
Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. Г( + х) dx '0
Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. (x² – 4x + 2)
Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. '5 r5 dx [ (4
Express the limit as a definite integral on the given interval. х? п [1, 3] lim 2 -Дх, - (х*)? + 4 п— 00 i=1
Express the limit as a definite integral on the given interval. п lim E [5(x)° – 4.x#]Ax, [ [2, 7] 4.x]Ax, i=1
Express the limit as a definite integral on the given interval. п lim E x; /1 + x} Ax, [2, 5] 3 Xị п> 00 j-1 i=1
Express the limit as a definite integral on the given interval. eti -Дх, [0, 1] п lim 2 п3 00 i=1
With a programmable calculator or computer (see the instructions for Exercise 5.1.9), compute the left and right Riemann sums for the function with n = 100. Explain why these estimates show that
Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. x sin?x dx, п 3 4 т
Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. х '2 dx, п 3D 5 Jо х + 1
Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. I Vx3 + 1 dx, n= 5
The table gives the values of a function obtained from an experiment. Use them to estimateUsing three equal subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints. If the
The graph of t is shown. Estimate with six sub-intervals using (a) right endpoints, (b) left endpoints, and (c) midpoints. 9(x) dx -2
(a) Find the Riemann sum for with four terms, taking the sample points to be right endpoints.(Give your answer correct to six decimal places.) Explain what the Riemann sum represents with the
If find the Riemann sum with n = 6, taking the sample points to be midpoints. What doesthe Riemann sum represent? Illustrate with a diagram. If f(x) = x² – 4,0 < x < 3,
Evaluate the Riemann sum for with five subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents. f(x) = x – 1, –6 < x
If A is the area under the curve y = ex from 1 to 3, use Exercise 27 to find a value of n such that Rn - A < 0.0001.
Determine a region whose area is equal to the given limit. Do not evaluate the limit. Зi 3 lim 2- п i-1 n
Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. f(x) = Vsin x, 0
Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. .2 f(x) = x² + /1 + 2x, 4
Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. 2x f(x) 1
In someone infected with measles, the virus level N (measured in number of infected cells per mL of blood plasma) reaches a peak density at about t = 12 days (when a rash appears) and then decreases

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