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study help
mathematics
precalculus
Questions and Answers of
Precalculus
a. Find the inverse of the function ƒ(x) = mx, where m is a constant different from zero.b. What can you conclude about the inverse of a function y = ƒ(x) whose graph is a line through the origin
Use the properties of logarithms to write the expression as a single term.a.b.c. In sin - In sin 0 5
a. Find the inverse of ƒ(x) = x + 1. Graph ƒ and its inverse together. Add the line y = x to your sketch, drawing it with dashes or dots for contrast.b. Find the inverse of ƒ(x) = x + b (b
Express the following logarithms in terms of ln 2 and ln 3.a. ln 0.75 b. ln (4/9)c. ln (1/2) d. ln 3√9e. ln 3√2 f. ln √13.5
Graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15–1.17, and applying an appropriate transformation.
Find simpler expressions for the quantitie.a. eln 7.2b. e-ln x2 c. eln x-ln y
Graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15–1.17, and applying an appropriate transformation.
Find simpler expressions for the quantitie.a. 2ln√e b. ln (ln ee) c. ln (e-x2-y2)
Solve for y in terms of t or x, as appropriate.ln y = 2t + 4
Suppose the graph of g is given. Write equations for the graphs that are obtained from the graph of g by shifting, scaling, or reflecting, as indicated.a. Up 1/2 unit, right 3b. Down 2 units, left
Solve for y in terms of t or x, as appropriate.ln (y - b) = 5t
Solve for y in terms of t or x, as appropriate.ln (y - 1) - ln 2 = x + ln x
Simplify the expression.a. 5log5 7b. 8log8√2 c. 1.3log1.3 75d. log4 16 e. log3√3f. log4 4
Solve for k.a. e2k = 4 b. 100e10k = 200 c. ek/1000 = a
Sketch the graph of the given function. What is the period of the function?y = cos 2x
Solve for t.a. e-0.3t = 27b. ekt = 1/2c. e(ln 0.2)t = 0.4
Sketch the graph of the given function. What is the period of the function?y = sin πx
Solve for t.e√t = x2
Express the ratios as ratios of natural logarithms and simplify.a.b.c. log₂x log3 x
ABC is a right triangle with the right angle at C. The sides opposite angles A, B, and C are a, b, and c, respectively.a. Find a and b if c = 2, B = π/3.b. Find a and c if b = 2, B = π/3.
Simplify the expression.a. 2log4x b. 9log3x c. log2 (e(ln 2)(sin x))
Find a formula for the inverse function ƒ -1 and verify that (ƒ ∘ ƒ-1 )(x) = (ƒ-1 ∘ ƒ)(x) = x.a.b. f(x) = 100 1 + 2x
ABC is a right triangle with the right angle at C. The sides opposite angles A, B, and C are a, b, and c, respectively.a. Express a in terms of B and b.b. Express c in terms of A and a.
Two wires stretch from the top T of a vertical pole to points B and C on the ground, where C is 10 m closer to the base of the pole than is B. If wire BT makes an angle of 35° with the horizontal
If ƒ(x) is one-to-one, can anything be said about g(x) = -ƒ(x)? Is it also one-to-one? Give reasons for your answer.
Suppose that the range of g lies in the domain of ƒ so that the composite ƒ ∘ g is defined. If ƒ and g are one-to-one, can anything be said about ƒ ∘ g? Give reasons for your answer.
The equation x2 = 2x has three solutions: x = 2, x = 4, and one other. Estimate the third solution as accurately as you can by graphing.
Use the graph shown in the figure. To print an enlarged copy of the graph, go to MathGraphs.com.Identify or sketch each of the quantities on the figure.(a) f(1) and f(4)(b) f(4) − f(1)(c) 4 −
Estimate the slope of the graph at the points (x1, y1) and (x2, y2). (x₁, y₁) y quate (²x²x), X almente
Use the graph to estimate the slope of the tangent line to y = xn at the point (1, 1). Verify your answer analytically. To print an enlarged copy of the graph, go to MathGraphs.com.(a)(b) y =
Describe how to find the slope of the tangent line to the graph of a function at a point.
Describe the Product Rule in your own words.
Describe how to find the derivative of a function using the limit process.
What are the derivatives of the sine and cosine function?
Use the Product Rule to find the derivative of the function.g(x) = (2x − 3)(1 − 5x)
Use the Product Rule to find the derivative of the function.h(t) = √t(1 − t2)
Find the slope of the tangent line to the graph of the function at the given point.f(x) = 3 − 5x, (−1, 8)
Use the Quotient Rule to find the derivative of the function. f(x) X x-5
Find the slope of the tangent line to the graph of the function at the given point.f(x) = 2x2 − 3, (2, 5)
Use the Quotient Rule to find the derivative of the function. g(x) sin x x²
Use the Quotient Rule to find the derivative of the function. h(x) = √x x³ + 1
Find f′(x) and f′(c).
Find f′(x) and f′(c). Function
Use the rules of differentiation to find the derivative of the function.f(x) = 9√x
Find the derivative of the function by the limit process. f(x) = 1 x - 1
Find the derivative of the function by the limit process.f(x) = −5x
Complete the table to find the derivative of the function. Original Function 2 7x4 y = Rewrite Differentiate Simplify
Complete the table to find the derivative of the function without using the Quotient Rule. Function y = x³ + 6x 3 Rewrite Differentiate Simplify
Find f′(x) and f′(c).Function Value of cf(x) = (x3 + 4x)(3x2 + 2x − 5)
Use the rules of differentiation to find the derivative of the function.f(t) = −3t2 + 2t − 4
Use the rules of differentiation to find the derivative of the function.g(x) = x2 + 4x3
Find the derivative of the algebraic function. g(s) = $³5 S s+2,
Find the derivative of the function by the limit process.f(x) = x3 − 12x
Use the rules of differentiation to find the derivative of the function.y = 2- sin
Find the derivative of the trigonometric function. y = 3(1 - sin x) 2 cos x
Find the derivative of the trigonometric function. f(t) COS t t
Use the rules of differentiation to find the derivative of the function.y = x2 − 1/2 cos x
The limit represents f′(c) for a function f and a number c. Find f and c. lim Δ.x - 0 [5 - 3(1 + Ax)] - 2 Δε
Find an equation of the line that is tangent to the graph of f and parallel to the given line.Function Linef(x) = −1/4 x2
The limit represents f′(c) for a function f and a number c. Find f and c. lim -x² + 36 9- x 9-x
Find the derivative of the function.f(x) = x2 + 5 − 3x−2
Find the derivative of the trigonometric function.f(t) = t2 sin t
Find equations of the two tangent lines to the graph of f that pass through the indicated point. f(x) = 4x - x² 5 4 3 نا 2 1 y (2,5) 1 2 3 5 X
Find the derivative of the trigonometric function.f(x) = −x + tan x
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.
Find the derivative of the trigonometric function.g(t) = 4√t + 6 csc t
Find the derivative of the function.f(x) = 6√x + 5 cos x
Do f and f′ always have the same domain? Explain.
(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 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
(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 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
Describe the x-values at which f is differentiable. f(x) = (x + 4)2/3 H -6 -4 H - 2 4 y -2 + X
Identify a function f that has the given characteristics. Then sketch the function.f(0) = 2; f′(x) = −3 for −∞ < x < ∞
Describe the x-values at which f is differentiable. f(x) = √x + 1 + 1 -2 4 3 نیا -1 y + 1 2 3
Find an equation of the tangent line to the graph at the given point. (The graphs in Exercises 69 and 70 are called Witches of Agnesi. The graphs in Exercises 71 and 72 are called serpentines.)
Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.y = x + sin x, 0 ≤ x < 2
Consider the function f(x) = x3/2 with the solution point (4, 8).(a) Use a graphing utility to graph f. Use the zoom feature to obtain successive magnifications of the graph in the neighborhood of
Consider the function f(x) = 1/2x2.(a) Use a graphing utility to graph the function and estimate the values of f′(0), f′(1/2), f′(1), and f′(2).(b) Use your results from part (a) to determine
The relationship between f and g is given. Explain the relationship between f′ and g′.g(x) = f(x) + 6
The table shows the national health care expenditures h (in billions of dollars) in the United States and the population p (in millions) of the United States for the years 2008 through 2013. The year
The relationship between f and g is given. Explain the relationship between f′ and g′.g(x) = −5 f(x)
Sketch the graph of a function f such that f′ > 0 for all x and the rate of change of the function is decreasing.
Use a graphing utility to graph the function and find the x values at which f is differentiable.f(x) = ∣x − 5∣
Find the second derivative of the function. f(x): X x-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 f′(x) = g′(x), then f(x) = g(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 tangent line to the differentiable function f at the
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 = 2, then dy/dx = 2.
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) = 0, then f′(x) is undefined.
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(t) = 3t + 5, [1,
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. ++ - 4 4 2 -2 -2 y 111 - X 4
A line with slope m passes through the point (0, 4) and has the equation y = mx + 4.(a) Write the distance d between the line and the point (3, 1) as a function of m.(b) Use a graphing utility to
Find the second derivative of the function.f(x) = 4x3/2
The graphs of f, f′, and f″ are shown on the same set of coordinate axes. Identify each graph. Explain your reasoning. To print an enlarged copy of the graph, go to MathGraphs.com. -2
Use the position function s(t) = −16t2 + v0t + s0 for free-falling objects.A silver dollar is dropped from the top of a building that is 1362 feet tall.(a) Determine the position and velocity
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 has derivatives from both the right and the left at a point, then
Find the second derivative of the function.f(x) = csc x
Find the given higher-order derivative.f′(x) = x3 − x2/5, f (3) (x)
Polynomials of what degree satisfy f (n) = 0? Explain your reasoning.
Where are the functions f1(x) = ∣sin x∣ and f2(x) = sin ∣x∣ differentiable?
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