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mathematics
precalculus

Questions and Answers of Precalculus

Give the position function s = ƒ(t) of an object moving along the s-axis as a function of time t. Graph ƒ together with the velocity function y(t) = ds/dt = ƒ′(t) and the acceleration function
Find the derivatives of the function.s = 2t3/2 + 3e2
Find the derivatives of the function.y = x-3/5 + π3/2
Find the derivatives of the function. r = eº 1 0² +0-π/2
Find the derivatives of the function. y = √x9.6 +2e¹.3
Find the derivatives of all orders of the function. y x4 2 3 -x² 2 - X 2
Find the derivatives of all orders of the function. y = x5 120
Give the position function s = ƒ(t) of an object moving along the s-axis as a function of time t. Graph ƒ together with the velocity function y(t) = ds/dt = ƒ′(t) and the acceleration function
Find the first and second derivatives of the function. W = 1 + 3z 3z (3 — z)
Do the graphs of the function have any horizontal tangents in the interval 0 ≤ x ≤ 2π? If so, where? If not, why not? Visualize your findings by graphing the functions with a grapher.y = x + sin
Find the first and second derivatives of the function. S t² + 5t - 1 +2
Do the graphs of the function have any horizontal tangents in the interval 0 ≤ x ≤ 2π? If so, where? If not, why not? Visualize your findings by graphing the functions with a grapher.y = 2x +
Find an equation for (a) The tangent to the curve at P and (b) The horizontal tangent to the curve at Q. 2 1 0 y Q 1 P 2 → X π 2 y = 4 + cotx - 2 csc x
Do the graphs of the function have any horizontal tangents in the interval 0 ≤ x ≤ 2π? If so, where? If not, why not? Visualize your findings by graphing the functions with a grapher.y = x + 2
Find the first and second derivatives of the function. r (0 − 1)(0² + 0 + 1) 03
Find the first and second derivatives of the function. И (x² + x)(x² − x + 1) x4
Find the first and second derivatives of the function. y x³ + 7 X
Find an equation for (a) The tangent to the curve at P (b) The horizontal tangent to the curve at Q. 4 y 0 P 4) 2 Q T1 4 y = 1 + √√√2 csc x + cotx 3 X
Do the graphs of the function have any horizontal tangents in the interval 0 ≤ x ≤ 2π? If so, where? If not, why not? Visualize your findings by graphing the functions with a grapher.y = x - cot
Find the first and second derivatives of the function. q² + 3 р (q − 1)³ + (q + 1)³
Find all points on the curve y = tan x, -π/2 < x < π/2, where the tangent line is parallel to the line y = 2x. Sketch the curve and tangent(s) together, labeling each with its equation.
Find the derivatives of all orders of the function.y = (x - 1) (x + 2)(x + 3)
Find all points on the curve y = cot x, 0 < x < π, where the tangent line is parallel to the line y = -x. Sketch the curve and tangent(s) together, labeling each with its equation.
Find the derivatives of all orders of the function.y = (4x2 + 3)(2 - x) x
Is there a value of b that will makecontinuous at x = 0? Differentiable at x = 0? Give reasons for your answers. g(x) = √x + b₂ x < 0 COS X, x ≥ 0
Find the first and second derivatives of the function.w = 3z2e2z
Is there a value of c that will makecontinuous at x = 0? Give reasons for your answer. f(x) = sin² 3x x² C, x = 0 x = 0
Find the tangents to Newton’s serpentine (graphed here) at the origin and the point (1, 2). y = 4x .2 x² + 1 y 2 1 0/ (1, 2) 1234 →X
The equations give the position s = ƒ(t) of a body moving on a coordinate line (s in meters, t in seconds). Find the body’s velocity, speed, acceleration, and jerk at time t = π/4 sec.s = 2 - 2
Find the tangent to the Witch of Agnesi (graphed here) at the point (2, 1). 1. 0 y = 8 x²+4 2 (2, 1) 1 2 3 X
Find the first and second derivatives of the function.w = ez(z - 1)(z2 + 1)
A weight is attached to a spring and reaches its equilibrium position (x = 0). It is then set in motion resulting in a displacement of x = 10 cos t, where x is measured in centimeters and t is
Suppose u and v are functions of x that are differentiable at x = 0 and that u(0) = 5, u′(0) = -3, v(0) = -1, v′(0) = 2.Find the values of the following derivatives at x = 0.a. d/dx(uv)b.
Suppose u and v are differentiable functions of x and that u(1) = 2, u′(1) = 0, v(1) = 5, v′(1) = -1.Find the values of the following derivatives at x = 0.a. d/dx(uv)b. d/dx(u/v)c. d/dx(v/u)d.
Graph y = cos x for -π ≤ x ≤ 2π. On the same screen, graphfor h = 1, 0.5, 0.3, and 0.1. Then, in a new window, try h = -1, -0.5, and -0.3. What happens as h → 0+? As h → 0-? What phenomenon
The equations give the position s = ƒ(t) of a body moving on a coordinate line (s in meters, t in seconds). Find the body’s velocity, speed, acceleration, and jerk at time t = π/4 sec.s = sin t +
a. Find equations for the horizontal tangents to the curve y = x3 - 3x - 2. Also find equations for the lines that are perpendicular to these tangents at the points of tangency.b. What is the
Find all points (x, y) on the graph of ƒ(x) = x2 with tangent lines passing through the point (3, 8). 10 6 2 -2 f(x) = x? (3,8) (x, y) 2 4 →
Graph y = -sin x for-π ≤ x ≤ 2π. On the same screen, graphfor h = 1, 0.5, 0.3, and 0.1. Then, in a new window, try h = -1, -0.5, and -0.3. What happens as h→ 0+? As h→ 0-? What phenomenon
Evaluate each limit by first converting each to a derivative at a particular x-value. lim X1 +50 X - 1 1
Evaluate each limit by first converting each to a derivative at a particular x-value. x2/9 x²/⁹ - 1 lim X1 X + 1
What happens to the derivatives of sin x and cos x if x is measured in degrees instead of radians? To find out, take the following steps.a. With your graphing calculator or computer grapher in degree
By computing the first few derivatives and looking for a pattern, find d999/dx999 (cos x).
The curve y = ax2 + bx + c passes through the point (1, 2) and is tangent to the line y = x at the origin. Find a, b, and c.
The curves y = x2 + ax + b and y = cx - x2 have a common tangent line at the point (1, 0). Find a, b, and c.
Find the values of a and b that make the following function differentiable for all x-values. Jax + b₂ bx²-3, f(x) = { x>-1 x≤-1
Derive the formula for the derivative with respect to x ofa. sec x. b. csc x. c. cot x.
Find all points (x, y) on the graph of ƒ(x) = 3x2 - 4x with tangent lines parallel to the line y = 8x + 5.
If gas in a cylinder is maintained at a constant temperature T, the pressure P is related to the volume V by a formula of the formin which a, b, n, and R are constants. Find dP/dV. P = nRT V
Find the value of a that makes the following function differentiable for all x-values. g(x) = (ax, x² 3x, - if x < 0 if x ≥ 0
Find all points (x, y) on the graph of g(x) = 1/3x3 - 3/2x2 + 1 with tangent lines parallel to the line 8x - 2y = 1.
Assume that a particle’s position on the x-axis is given by x = 3 cos t + 4 sin t, where x is measured in feet and t is measured in seconds.a. Find the particle’s position when t = 0, t = π/2,
Find all points (x, y) on the graph of y = x/(x - 2) with tangent lines perpendicular to the line y = 2x + 3.
Use the Derivative Quotient Rule to prove the Power Rule for negative integers, that is,where m is a positive integer. d dx (_X) || -mx-m-1
The general polynomial of degree n has the formwhere an ≠ 0. Find P′(x). P(x) = ax + a₁-1.xn-1 + + a₂x² + a₁x + ao
a. The Reciprocal Rule says that at any point where the function v(x) is differentiable and different from zero,Show that the Reciprocal Rule is a special case of the Derivative Quotient Rule.b. Show
Graph y = tan x and its derivative together on (-π/2, π/2). Does the graph of the tangent function appear to have a smallest slope? A largest slope? Is the slope ever negative? Give reasons for
Graph y = cot x and its derivative together for 0 < x < π. Does the graph of the cotangent function appear to have a smallest slope? A largest slope? Is the slope ever positive? Give reasons
Graph y = (sin x)/x, y = (sin 2x)/x, and y = (sin 4x)/x together over the interval -2 ≤ x ≤ 2. Where does each graph appear to cross the y-axis? Do the graphs really intersect the axis? What
Suppose that the function v in the Derivative Product Rule has a constant value c. What does the Derivative Product Rule then say? What does this say about the Derivative Constant Multiple Rule?
One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise iswhere q is the quantity you order when things run low (shoes, TVs,
Using the definition, calculate the derivatives of the function. Then find the values of the derivatives as specified. g(t) g'(−1), g'(2), g'(√3) =
The equation ax2 + 2x - 1 = 0, where a is a constant, has two roots if a > -1 and a ≠ 0, one positive and one negative:a. What happens to r+(a) as a→ 0? As a→ -1+?b. What happens to r-(a) as
Using the definition, calculate the derivatives of the function. Then find the values of the derivatives as specified. r(s) = √2s + 1; r'(0), r'(1), r'(1/2)
Find the indicated derivative. dy dx if y = 2x³
Find the indicated derivative. dr ds if r = $³ E + 22- -
Find the indicated derivative. ds dt if s t 2t + 1
Find the indicated derivative. dp 而 dq if p =q³/2
Find the indicated derivative. dz. dw if z = 1 Vwe - 1
Differentiate the function and find the slope of the tangent line at the given value of the independent variable. 9 f(x) = x + = X' x = -3
Find the indicated derivative. du dt if v = t 1 t
Differentiate the function and find the slope of the tangent line at the given value of the independent variable. y = x + 3 1 - x' x = -2
Differentiate the function. Then find an equation of the tangent line at the indicated point on the graph of the function. y = f(x) = 8 √x - 2 (x, y) = (6,4)
a. The graph in the accompanying figure is made of line segments joined end to end. At which points of the interval [-4, 6] is ƒ′ not defined? Give reasons for your answer.b. Graph the derivative
Differentiate the function and find the slope of the tangent line at the given value of the independent variable. k(x) 1 2 + x² x = 2
Using the definition, calculate the derivatives of the function. Then find the values of the derivatives as specified.ƒ(x) = 4 - x2; ƒ′(-3), ƒ′(0), ƒ′(1)
Using the definition, calculate the derivatives of the function. Then find the values of the derivatives as specified.F(x) = (x - 1)2 + 1; F′(-1), F′(0), F′(2)
Differentiate the function. Then find an equation of the tangent line at the indicated point on the graph of the function. W = = g(z) = 1 + V4 - z, (z, w) = (3, 2)
Find the values of the derivative. dy dx x= √3 if y = 1 - 1 X
Find the values of the derivative. dr de 0=0 if r 2 V4 - 0
Compute the right-hand and left-hand derivatives as limits to show that the function are not differentiable at the point P. y=x² P(0, 0) y y = f(x) y = x X
Find the values of the derivative. dw dz |z=4 if w z + Vz =
Compute the right-hand and left-hand derivatives as limits to show that the function are not differentiable at the point P. y = 2 y 2 1 0 y = f(x) P(1, 2) 1 y = 2x 2 X
a. Use the following information to graph the function ƒ over the closed interval [-2, 5].i) The graph of ƒ is made of closed line segments joined end to end.ii) The graph starts at the point (-2,
Populations starting out in closed environments grow slowly at first, when there are relatively few members, then more rapidly as the number of reproducing individuals increases and resources are
Compute the right-hand and left-hand derivatives as limits to show that the function are not differentiable at the point P. y 1 0 y = f(x) y = 2x - 1 P(1, 1) y = √x 1 X
Jared Fogle, also known as the “Subway Sandwich Guy,” weighed 425 lb in 1997 before losing more than 240 lb in 12 months (http://en.wikipedia.org/wiki/Jared_Fogle). A chart showing his possible
Differentiate the function and find the slope of the tangent line at the given value of the independent variable.s = t3 - t2, t = -1
Determine if the piecewise-defined function is differentiable at the origin. f(x): = (2x - 1, x² + 2x + 7₂ 7, x ≥ 0 x < 0
Shows the graph of a function over a closed interval D. At what domain points does the function appear to bea. Differentiable?b. Continuous but not differentiable?c. Neither continuous nor
Compute the right-hand and left-hand derivatives as limits to show that the function are not differentiable at the point P. y = x - y = f(x) P(1, 1) 1 y X X
Determine if the piecewise-defined function is differentiable at the origin. g(x) = x > [x2/3, x = 0 (1/3 x < 0
Shows the graph of a function over a closed interval D. At what domain points does the function appear to bea. Differentiable?b. Continuous but not differentiable?c. Neither continuous nor
Shows the graph of a function over a closed interval D. At what domain points does the function appear to bea. Differentiable?b. Continuous but not differentiable?c. Neither continuous nor
Shows the graph of a function over a closed interval D. At what domain points does the function appear to bea. Differentiable?b. Continuous but not differentiable?c. Neither continuous nor
Shows the graph of a function over a closed interval D. At what domain points does the function appear to bea. Differentiable?b. Continuous but not differentiable?c. Neither continuous nor
Shows the graph of a function over a closed interval D. At what domain points does the function appear to bea. Differentiable?b. Continuous but not differentiable?c. Neither continuous nor
a. Let ƒ(x) be a function satisfying |ƒ(x)| ≤ x2 for -1 ≤ x ≤ 1. Show that ƒ is differentiable at x = 0 and find ƒ′(0).b. Show thatis differentiable at x = 0 and find ƒ′(0).
Graph y = 1/12√x2 in a window that has 0 ≤ x ≤ 2. Then, on the same screen, graphfor h = 1, 0.5, 0.1. Then try h = -1, -0.5, -0.1. Explain what is going on. y = √x + h = √x 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

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