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study help
mathematics
precalculus
Questions and Answers of
Precalculus
Graph y = 3x2 in a window that has -2 ≤ x ≤ 2, 0 ≤ y ≤ 3. Then, on the same screen, graph y = (x + h)³ - x³ h
a. Find the derivative ƒ′(x) of the given function y = ƒ(x).b. Graph y = ƒ(x) and y = ƒ′(x) side by side using separate sets of coordinate axes, and answer the following questions.c. For what
a. Find the derivative ƒ′(x) of the given function y = ƒ(x).b. Graph y = ƒ(x) and y = ƒ′(x) side by side using separate sets of coordinate axes, and answer the following questions.c. For what
a. Find the derivative ƒ′(x) of the given function y = ƒ(x).b. Graph y = ƒ(x) and y = ƒ′(x) side by side using separate sets of coordinate axes, and answer the following questions.c. For what
Does the parabola y = 2x2 - 13x + 5 have a tangent whose slope is -1? If so, find an equation for the line and the point of tangency. If not, why not?
Does any tangent to the curve y = √x cross the x-axis at x = -1? If so, find an equation for the line and the point of tangency. If not, why not?
Does knowing that a function ƒ(x) is differentiable at x = x0 tell you anything about the differentiability of the function -ƒ at x = x0? Give reasons for your answer.
Does knowing that a function g(t) is differentiable at t = 7 tell you anything about the differentiability of the function 3g at t = 7? Give reasons for your answer.
Suppose that functions g(t) and h(t) are defined for all values of t and g(0) = h(0) = 0. Can limt→0 (g(t))/(h(t)) exist? If it does exist, must it equal zero? Give reasons for your answers.
The sum of the first eight terms of the Weierstrass functionGraph this sum. Zoom in several times. How wiggly and bumpy is this graph? Specify a viewing window in which the displayed portion of the
Use a CAS to perform the following steps for the function.a. Plot y = ƒ(x) over the interval (x0 - 1/2) ≤ x ≤ (x0 + 3).b. Holding x0 fixed, the difference quotientat x0 becomes a function of the
Find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together. y = 2√x, (1, 2)
Find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together. y = x² X (-1, 1)
Find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together.y = 4 - x2, (-1, 3)
Find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there. g(x) = X x - 2' (3, 3)
Find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together. y = 1 x3: -2, 1 8
Find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together.y = (x - 1)2 + 1, (1, 1)
Find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there. g(x) 8 +2² X (2, 2)
Find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together.y = x3, (-2, -8)
Find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there. f(x) = √x + 1, (8,3)
Find the slope of the curve at the point indicated. y 1 X - 1' x = 3
Find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.ƒ(x) = x2 + 1, (2, 5)
Find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.ƒ(x) = x - 2x2, (1,-1)
In a controlled laboratory experiment, yeast cells are grown in an automated cell culture system that counts the number P of cells present at hourly intervals. The number after t hours is shown in
Find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.ƒ(x) = √x, (4, 2)
Find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.h(t) = t3, (2, 8)
Find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.h(t) = t3 + 3t, (1, 4)
Find the slope of the curve at the point indicated.y = 5x - 3x2, x = 1
Find the slope of the curve at the point indicated.y = x3 - 2x + 7, x = -2
Find the slope of the curve at the point indicated. y = x 1 - x + 1' - x = 0
On a scale from 0 to 1, the effectiveness E of a pain-killing drug t hours after entering the bloodstream is displayed in the accompanying figure.a. At what times does the effectiveness appear to be
Does the graph ofhave a tangent at the origin? Give reasons for your answer.We say that a continuous curve y = ƒ(x) has a vertical tangent at the point where x = x0 if the limit of the difference
At what points do the graphs of the function have horizontal tangents?ƒ(x) = x2 + 4x - 1
At what points do the graphs of the function have horizontal tangents?g(x) = x3 - 3x
Does the graph ofhave a vertical tangent at the origin? Give reasons for your answer. f(x) = -1, 0, 1, x < 0 x = 0 x > 0
Does the graph ofhave a vertical tangent at the point (0, 1)? Give reasons for your answer. U(x) = Jo, 0, [1, x < 0 x ≥ 0
Does the graph ofhave a tangent at the origin? Give reasons for your answer. f(x) = 0 x '(x/1)UIS -x] 10, x = 0
Find equations of all lines having slope -1 that are tangent to the curve y = 1/(x - 1).
Find an equation of the straight line having slope 1/4 that is tangent to the curve y = √x.
An object is dropped from the top of a 100-m-high tower. Its height above ground after t sec is 100 - 4.9t2 m. How fast is it falling 2 sec after it is dropped?
At t sec after liftoff, the height of a rocket is 3t2 ft. How fast is the rocket climbing 10 sec after liftoff?
What is the rate of change of the area of a circle (A = πr2) with respect to the radius when the radius is r = 3?
What is the rate of change of the volume of a ball (V = (4/3)πr3) with respect to the radius when the radius is r = 2?
Show that the line y = mx + b is its own tangent line at any point (x0, mx0 + b).
Find the slope of the tangent to the curve y = 1/2x at the point where x = 4.
Graph the curve.a. Where do the graphs appear to have vertical tangents?b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction. y = √|4 - x|
Graph the curve.a. Where do the graphs appear to have vertical tangents?b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction.y = x2/5
Use a CAS to perform the following steps for the function.a. Plot y = ƒ(x) over the interval (x0 - 1/2) ≤ x ≤ (x0 + 3).b. Holding x0 fixed, the difference quotientat x0 becomes a function of the
Graph the curve.a. Where do the graphs appear to have vertical tangents?b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction.y = x4/5
Graph the curve.a. Where do the graphs appear to have vertical tangents?b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction.
Graph the curve.a. Where do the graphs appear to have vertical tangents?b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction.y = x1/5
Use a CAS to perform the following steps for the function.a. Plot y = ƒ(x) over the interval (x0 - 1/2) ≤ x ≤ (x0 + 3).b. Holding x0 fixed, the difference quotientat x0 becomes a function of the
Graph the curve.a. Where do the graphs appear to have vertical tangents?b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction.y = x3/5
Graph the curve.a. Where do the graphs appear to have vertical tangents?b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction.y = 4x2/5 - 2x
Graph the curve.a. Where do the graphs appear to have vertical tangents?b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction.y = x5/3 - 5x2/3
Graph the curve.a. Where do the graphs appear to have vertical tangents?b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction.y = x2/3 - (x - 1)1/3
Graph the curve.a. Where do the graphs appear to have vertical tangents?b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction.y = x1/3 + (x - 1)1/3
Use a CAS to perform the following steps for the function.a. Plot y = ƒ(x) over the interval (x0 - 1/2) ≤ x ≤ (x0 + 3).b. Holding x0 fixed, the difference quotientat x0 becomes a function of the
Graph the functionThen discuss, in detail, limits, one-sided limits, continuity, and one-sided continuity of ƒ at x = -1, 0, and 1. Are any of the discontinuities removable? Explain. f(x)
Say whether the function graphed is continuous on [-1, 3]. If not, where does it fail to be continuous and why? - 2 1 0 y y = f(x) 2 3 X
As the number x increases through positive values, the numbers 1/x and 1 / (ln x) both approach zero. What happens to the numberas x increases? Here are two ways to find out.a. Evaluate ƒ for x =
Repeat the instructions of Exercise 1 forGraph the functionThen discuss, in detail, limits, one-sided limits, continuity, and one-sided continuity of ƒ at x = -1, 0, and 1. Are any of the
Find the limit of each function (a) As x→ ∞ and (b) As x→ -∞. f(x) = // - 3
In relativity theory, the length of an object, say a rocket, appears to an observer to depend on the speed at which the object is traveling with respect to the observer. If the observer measures the
Give an informal or intuitive definition of the limitWhy is the definition “informal”? Give examples. lim f(x) = L. X-C
Find the limit of each function (a) As x→ ∞ and (b) As x→ -∞. g(x) || 1 2 + (1/x)
Find the limit of each function (a) As x→ ∞ and (b) As x→ -∞. f(x) = = 7T TT 2
Torricelli’s law says that if you drain a tank like the one in the figure shown, the rate y at which water runs out is a constant times the square root of the water’s depth x. The constant
The rules of exponents tell us that a0 = 1 if a is any number different from zero. They also tell us that 0n = 0 if n is any positive number.If we tried to extend these rules to include the case 00,
What is the average rate of change of the function g(t) over the interval from t = a to t = b? How is it related to a secant line?
Find the limit of each function (a) As x→ ∞ and (b) As x→ -∞. g(x) 11 1 8 - (5/x²)
The interior of a typical 1-L measuring cup is a right circular cylinder of radius 6 cm. The volume of water we put in the cup is therefore a function of the level h to which the cup is filled, the
What limit must be calculated to find the rate of change of a function g(t) at t = t0?
Find the limit of each function (a) as x→ ∞ and (b) as x→ -∞. h(x) = −5+ (7/x) 3 - (1/x²)
Does the existence and value of the limit of a function ƒ(x) as x approaches c ever depend on what happens at x = c? Explain and give examples.
As you may know, most metals expand when heated and contract when cooled. The dimensions of a piece of laboratory equipment are sometimes so critical that the shop where the equipment is made must be
What function behaviors might occur for which the limit may fail to exist? Give examples.
What theorems are available for calculating limits? Give examples of how the theorems are used.
Find the limit of each function (a) As x→ ∞ and (b) As x→ -∞. h(x) = 3 - (2/x) 4 + (√2/x²)
Use the method in Example 3 to find (a) The slope of the curve at the given point P, and (b) An equation of the tangent line at P.y = x2 - 5, P(2, -1)Example 3Find the slope of the parabola
Use the formal definition of limit to prove that the function is continuous at c.ƒ(x) = x2 - 7, c = 1
How are one-sided limits related to limits? How can this relationship sometimes be used to calculate a limit or prove it does not exist? Give examples.
Use the method in Example 3 to find (a) The slope of the curve at the given point P, and (b) An equation of the tangent line at P.y = 7 - x2, P(2, 3)Example 3Find the slope of the parabola
Use the formal definition of limit to prove that the function is continuous at c.g(x) = 1/(2x), c = 1/4
What is the value of limθ→0 ((sin θ)/θ)? Does it matter whether θ is measured in degrees or radians? Explain.
Use the method in Example 3 to find (a) The slope of the curve at the given point P, and (b) An equation of the tangent line at P.y = x2 - 2x - 3, P(2, -3)Example 3Find the slope of the
Use the formal definition of limit to prove that the function is continuous at c.h(x) = √2x - 3, c = 2
What exactly does limx→c ƒ(x) = L mean? Give an example in which you find a δ > 0 for a given ƒ, L, c, and ϵ > 0 in the precise definition of limit.
Use the method in Example 3 to find (a) The slope of the curve at the given point P, and (b) An equation of the tangent line at P.y = x2 - 4x, P(1, -3)Example 3Find the slope of the
Use the formal definition of limit to prove that the function is continuous at c.F(x) = √9 - x, c = 5
Use the method in Example 3 to find (a) The slope of the curve at the given point P, and (b) An equation of the tangent line at P.y = x3, P(2, 8)Example 3Find the slope of the parabola y =
Show that a function cannot have two different limits at the same point. That is, if limx→c ƒ(x) = L1 and limx→c ƒ(x) = L2, then L1 = L2.
What conditions must be satisfied by a function if it is to be continuous at an interior point of its domain? At an endpoint?
Use the method in Example 3 to find (a) The slope of the curve at the given point P, and (b) An equation of the tangent line at P.y = 2 - x3, P(1, 1)Example 3Find the slope of the parabola
Find the limit of each rational function (a) As x→ ∞ and (b) As x→ -∞. h(x) = 7x3 x3 x³ - 3x² + 6x
Suppose limx→0 ƒ(x) = 1 and limx→0 g(x) = -5. Name the rules in Theorem 1 that are used to accomplish steps (a), (b), and (c) of the following calculation.a. b. c. 2f(x) - g(x) lim x-0 (f(x)
Find the limit of each rational function (a) As x→ ∞ and (b) As x→ -∞. f(x) = 2x + 3 - 5x +7
Prove the limit Constant Multiple Rule:for any constant k. lim kf(x) = k lim f(x) X-C X-C
Find the limit of each rational function (a) As x→ ∞ and (b) As x→ -∞. f(x) 2x³ + 7 x3 x³ = x² + x + 7
Use the formal definition of limit to prove that the function has a continuous extension to the given value of x. f(x) x² - 1 x + 1' I- X x = -1
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