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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

Fill in the following table with either even or odd and prove each result. Assume n is a nonnegative integer and fn means the nth power of f. n is n is odd even f is even f" is f" is f is odd f"
Simplify the following expressions. d. J. dx Vt4 + 1 dt
Fill in the blanks with right, left, or midpoint; an interval; and a value of n. In some cases, more than one answer may work. k (1.5+)-sa 2 k=1 interval [______, _] with n = Riemann sum for f on the
Simplify the following expressions. Г dx 2 d dp p2
For what functions f is it true that the square of the average value of f equals the average value of the square of f over all intervals [a, b]?
Use a change of variables to evaluate the following integrals. 8) √x dx (x³/2+8
Simplify the following expressions. d #f dx -x V1 + 1² dt
Consider the function f and its graph.a. Estimate the zeros of the area function b. Estimate the points (if any) at which A has a local maximum or minimum.c. Sketch a graph of A, for 0 ≤ x ≤ 10,
Use a change of variables to evaluate the following integrals. [² sin x sec8 x dx
Simplify the following expressions. f dx dp 2 p² + 1
Consider the function f and its graph.a. Estimate the zeros of the area functionb. Estimate the points (if any) at which A has a local maximum or minimum.c. Sketch a graph of A, for 0 ≤ x ≤ 10,
Simplify the following expressions. = |² dx e²x In t² dt
Use a change of variables to evaluate the following integrals. 3 x = dx 2 √x² - 1 2
Use a change of variables to evaluate the following integrals. In x X -dx
Consider the function f and its graph.a. Estimate the zeros of the area function b. Estimate the points (if any) at which A has a local maximum or minimum.c. Sketch a graph of A, for 0 ≤ x ≤ 10,
Use geometry to evaluate the following integrals. 3 |x + 1| dx -2
Use a change of variables to evaluate the following integrals. 6/5 0 dx 25x² + 36
Use geometry to evaluate the following integrals. Si |2x - 4 dx
Consider the function f and its graph.a. Estimate the zeros of the area function b. Estimate the points (if any) at which A has a local maximum or minimum.c. Sketch a graph of A, for 0 ≤ x ≤ 10,
Use a change of variables to evaluate the following integrals. x³ V16 x4 dx -
Use geometry to evaluate the following integrals. S (3x - 6) dx
Consider the function f and the points a, b, and c.a. Find the area function using the Fundamental Theorem.b. Graph f and A.c. Evaluate A(b) and A(c) and interpret the results using the graphs of
Consider the function f and the points a, b, and c.a. Find the area function using the Fundamental Theorem.b. Graph f and A.c. Evaluate A(b) and A(c) and interpret the results using the graphs of
Use a change of variables to evaluate the following integrals. V3 Sv (x - 1)(x² - 2x)¹¹ dx ܗܘܢ
Consider the function f and the points a, b, and c.a. Find the area function using the Fundamental Theorem.b. Graph f and A.c. Evaluate A(b) and A(c) and interpret the results using the graphs of
Use geometry to evaluate the following integrals. L √24 - 2x - x² dx -6
Consider the function f and the points a, b, and c.a. Find the area function using the Fundamental Theorem.b. Graph f and A.c. Evaluate A(b) and A(c) and interpret the results using the graphs of
Use a change of variables to evaluate the following integrals. S 0 (x + 1)(x + 2) 2x³ + 9x² + 12x + 3 2 - dx 36
Consider the function f and the points a, b, and c.a. Find the area function using the Fundamental Theorem.b. Graph f and A.c. Evaluate A(b) and A(c) and interpret the results using the graphs of
Use a change of variables to evaluate the following integrals. sin x - dx 2 + cos x
a. Multiply the numerator and denominator of sec x by sec x + tan x; then use a change of variables to show thatb. Show that [sec x sec x dx = ln |sec x + tan x| + C.
Use a change of variables to evaluate the following integrals. 0 TT/4 esin² x sin 2 x dx
Use a change of variables to evaluate the following integrals. S 4 9x² + 6x + 1 dx
a. Use a change of variables to show thatb. Show that Stan tan x dx = -In cos x + C = In secx + C.
Simplify the given expressions. [r f'(t) dt, where f' is continuous on [3, 8]
Simplify the given expressions. d dx 0 dt 1² + 4 2
Simplify the given expressions. d dx Jo cos x (14 + 6) dt
Simplify the given expressions. d à fe dx e dt
What value of b > -1 maximizes the integral [₁x² (3- -1 x² (3x) dx?
Graph the function f(x) = 8 + 2x - x2 and determine the values of a and b that maximize the value of the integral T.C (8 + 2x - x²) dx.
For a given function ƒ, what does ƒ' represent?
Use Theorem 3.11 to evaluate the following limits. THEOREM 3.11 Trigonometric Limits sin x lim X-0 X 1 lim x-0 COS X x = 0
Find the derivative of the following functions.p(x) = 8x
Find dy/dx.y = ex√x3
Suppose ƒ(x) →100 and g(x) → 0, with g(x) Determine f(x) lim x-2 g(x)
Find the critical points of ƒ. Assume a is a constant. f(x)=x²-3ax² + 3a²x-a²
Find the critical points of ƒ. Assume a is a constant. 1 (x) = -√x³² - a²x 5
Find the critical points of ƒ. Assume a is a constant.ƒ(x) = x √x - a
Leta. Make two tables, one showing the values of g for t = 8.9, 8.99, and 8.999 and one showing values of g for t = 9.1, 9.01, and 9.001.b. Make a conjecture about the value of g(t) t - 9 Vt - 3*
Given the polynomialprove thatfor any value of a. +... + be + bạn p(x) = bx" + b₁-₁x²-1 +
Why is it not possible to evaluateby direct substitution? lim sin x x 0x
How is used in this section? sin x lim x-0 X >0 X
Identify the inner and outer functions in the composition cos4 x.
Suppose ƒ is a one-to-one function with f(2) = 8 and ƒ'(2) = 4. What is the value of (ƒ-1)'(8)?
Identify the inner and outer functions in the composition (x2 + 10)-5.
Explain why bx = ex ln b.
Find the derivative of the following functions.ƒ(x) = 3x4(2x2 - 1)
Use Theorem 3.11 to evaluate the following limits. THEOREM 3.11 Trigonometric sin x lim X-0 X Limits 1 lim X-0 COS X 1 = 0
Explain in your own words the adage: The average describes the past; the marginal describes the future.
Evaluate the derivatives of the following functions.ƒ(x) = sin-1 2x
Why is the notation dy/dx used to represent the derivative?
Use Theorem 3.11 to evaluate the following limits. THEOREM 3.11 Trigonometric Limits sin x lim X-0 X 1 lim x-0 COS X x = 0
Use Theorem 3.11 to evaluate the following limits. THEOREM 3.11 Trigonometric Limits sin x lim X-0 X 1 lim x-0 COS X x = 0
Use implicit differentiation to find dy/dx.sin xy = x + y
Use the definition of the derivative to do the following. Verify that g'(x) || 1 V2x - 3 where g(x) √2x - 3.
Use Theorem 3.11 to evaluate the following limits. THEOREM 3.11 Trigonometric Limits sin x lim X-0 X 1 lim x-0 COS X x = 0
Find the derivative of the following functions.g(w) = ew(w3 - 1)
Use Theorem 3.11 to evaluate the following limits. THEOREM 3.11 Trigonometric Limits sin x lim X-0 X 1 lim x-0 COS X x = 0
Evaluate the derivatives of the following functions.ƒ(x) = sin-1 (esin x)
Find the derivative of the following functions.f (v) = v100
Evaluate the derivatives of the following functions.ƒ(x) = sin-1 (e-2x)
Find the derivative of the following functions.h(t) = t
Find the derivative of the following functions.h(x) = (x - 1)(x3 + x2 + x + 1)
Evaluate the derivatives of the following functions.ƒ(x) = sin-1 (ln x)
Use the definition of the derivative to do the following.Verify that f'(x) = 4x - 3, where f(x) = 2x2 - 3x + 1.
Find the derivative of the following functions.g(w) = ew(5w2 + 3w + 1)
Evaluate the derivatives of the following functions.ƒ(w) = cos (sin-1 2w)
Find the derivative of the following functions.ƒ(t) = t5et
Evaluate the derivatives of the following functions.ƒ(x) = x sin-1 x
Find the derivative of the following functions.ƒ(x) = 5
Find the derivative of the following functions.g(x) = 6x - 2xex
Evaluate and simplify the following derivatives. d dx (5x (5x + sin³x + sin x³)
Use Theorem 3.11 to evaluate the following limits. THEOREM 3.11 Trigonometric Limits sin x lim X-0 X 1 lim x-0 COS X x = 0
Use Theorem 3.11 to evaluate the following limits. THEOREM 3.11 Trigonometric Limits sin x lim X-0 X 1 lim x-0 COS X x = 0
Evaluate and simplify the following derivatives. d dt (5t² sin t)
Use Theorem 3.11 to evaluate the following limits. THEOREM 3.11 Trigonometric Limits sin x lim X-0 X 1 lim x-0 COS X x = 0
Evaluate and simplify the following derivatives. d dx -(2xVx² 2x + 2)
Use Theorem 3.11 to evaluate the following limits. THEOREM 3.11 Trigonometric Limits sin x lim X-0 X 1 lim x-0 COS X x = 0
Evaluate and simplify the following derivatives. d dx x²= لیا 7x² + 7x + 1) + π.χ.
Evaluate the derivatives of the following functions.ƒ(z) = cot-1 √z
Use implicit differentiation to find dy/dx.cos y2 + x = ey
Find dy/dx for the following functions.y = sin x + cos x
Evaluate the derivatives of the following functions.g(z) = tan-1 (1/z)
Use implicit differentiation to find dy/dx.x + 2y = √y
Evaluate the derivatives of the following functions.ƒ(y) = tan-1 (2y2 - 4)
Use implicit differentiation to find dy/dx.exy = 2y
Evaluate the derivatives of the following functions.ƒ(x) = tan-1 10x
Find the derivative of the following functions.g(w)= 5/6 w12
Evaluate the derivatives of the following functions.ƒ(x) = x cot-1 (x/3)
Evaluate the derivatives of the following functions.ƒ(x) = csc-1 (tan ex)

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