calc

Given q(x)=(4x+3)/2x^2−1, find the critical points.

-1 and -1/2

Find the derivative of h(x)=cos^3 (x).

-3cos^2(x)sin(x)

Let h(x)=f(x)/g(x), where f(−7)=−4,f′(−7)=8,g(−7)=−2, and g′(−7)=−1. What is h′(−7)?

-5

Evaluate the limit. limx→0 x^2/x+6+(36/x−6)

-6

compute: limit as x approaches 0 (|4x-1|-|4x+1|)/x)

-8

Find the equation of the tangent line to the function f(x)=−2x^3−4x^2−3x+2 at the point where x=−1. Give your answer in the form y=mx+b.

-x+2

r(x)=(lnx^4)/2x What is limx→+∞ r(x)?

0

Given that f(2) = 0, g(2) = 0, f′(2) = 5, and g′(2)=3, evaluate limx→2 xf(x)/g(x):

10/3

Suppose k(x)=f(g(h(x))). Given the table of values below, determine k′(1). IN A GRAPH x: 134 f(x):−6−3−2 g(x):−347 h(x):313 f′(x):6−71 g′(x):−6−2−7 h′(x):−65−8

12

Given two nonnegative numbers x and y such that x+y=8, what is the difference between the maximum and minimum of the quantity x^2y^2/2?

128

Let h(x)=f(x)g(x). If f(x)=−4x^2+4x−5, g(2)=3, and g′(2)=−4, what is h′(2)?

16

A continuous function f(x) defined over the interval [0,2]verifies f(0) = −2 and f′(x) ≥2x for all 0< x ≤2. What's the minimum value possible for f(2)?

2

Evaluate the limit: limx→1 sin(2x−2)/x−1

2

Find the following limit. limx→9 of x−9/sq root of (x-3)-sq root of 6

2 sq root 6

Determine p′(x) if p(x)=(5x)^7x and x=4.

20^28(7ln20+7)

let f(x) = xe^(x^3-8) find f'(2)

25

Given the following piecewise function, evaluate limx→−4+f(x). f(x)=−2x2−x−3 2x^2+x+3 2x2−2 if x<−4 if −4≤x≤−2 if x>−2

31

The function f(x) is continuous on the interval [1,10]. The table below gives some of its values. What is the minimum number of zeros that f(x) is guaranteed to have by the Intermediate Value Theorem? a graph shows x values 1-10 and f(x) values -4, 4, 1, 2, -5, -3, 8, 6, 8, -4

4 zeroes

Evaluate the limit below. limx→0 7x/sin(3x)

7/3

determine the value of k so that the piecewise function below is continuous.f(x)={kx−1 k+5x. if x≤3 if x>3

8

A rectangle has a length that is increasing at a rate of 10 mm per second with the width being held constant. What is the rate of change of the area of the rectangle if the width is 8 mm? (Do not include the units in your answer.)

80

A rectangle with edges of length a and b is evolving in time through the relation a^2+a = lnb. The rate of change of a is kept constant at 2cm/s. At what rate is the area of the rectangle changing when a=1cm?

8e^2

Given that limx→−2 of f(x)=−4 and limx→−2 of g(x)=5, evaluate the following. limx→−2 of (f(x)−4g(x))

lim as x approaches -2 of (f(x)-4g(x))= -24

Given the following limit lim h→0 f(3+h)−f(3)/h. =1 which of the following must be true? Select two correct answers.

lim as x approaches 3 of f(x)=f(3) and f(x) is continuous at x=3

Let a>0 be a positive number. Compute: limn→∞ n((a^1/n)−1))

ln a

Using the First Derivative Test, what are the local extrema for the function s(x)=−2/3x^3+10x^2−48x−1?

local min x=4 local max at x=6

Determine limx→0+ (8/sinx)−(9/x).

neg infinity

what is r'(x) when r(x)=ln(x^(4/5)sinx/4x^2+x+3

r'(x)=4/5x + cosx/sinx - 8x+1/4x^2+x+3

Determine r′(x) when r(x)=0.09e^xcos(x).

r′(x)=0.09((e^x)(cos(x))+(e^x)(−sin(x))).

Is it possible to find a non-negative number a ≥0 such that the function f(x) = x+1 if x≤a and f(x) = x^2 if x > a is continuous at all points x?

the only non negative number that makes f(x) continuous is a=(1+ sq root 5)/2

A UCF math professor has sampled a function along with its derivatives: f(0) = 2, f(1) = 3, f(2) = −2 f′(0) = 1, f′(1) = 0, f′(2) = −1 f″(0) = −1, f″(1) = −0.25, f″(2) = 3 The function is defined for all x and has fist and second derivatives continuous for all points. Based on these data, choose ALL the correct answers:

there are points near zero such that f(x)>2 and the function f(x) has a local max at x=1

A UCF math professor has sampled a function along with its derivatives: f(0) = 2, f(1) = 3, f(2) = −2 f′(0) = 1, f′(1) = 0, f′(2) = −2 f″(0) = −1, f″(1) = −0.25, f″(2) = 3 The function is defined for all x≥0. Based on these data, choose ALL the correct answers:

there must exist a number 1<y<2 such that f'(y) = -5 AND there must exist a number 1<z<2 such that f"(z)=neg sq root 2

Let a>0 be a positive number. What's the minimum possible value of x^a ⋅lnx when 0<x≤1?

-1/(a*e)

q(x)=10sin(πx)/ln(x9) What is limx→1 q(x)?

-10/9 pi

If g(−1)=5, g(3)=−8, t(−1)=−5, and g′(x)=t′(x) on [−1,3], determine the value of t(3)?

-18

Find the derivative of y=(4x−2/2x−1)^3

0

Let f(x) and g(x)be functions with continuous 1st derivatives satisfying f(0) = g(0) = 0. Assume further that f′(0) = 3 & g′(0) = −4. Compute limx→0 (f(g(x))//g(f(x))).

1

If limx→1[f(x) + g(x)] = −2 and limx→1 [f(x) − g(x)] = 1, is it possible to compute limx→1 f(x)/g(x)?

1/3

Given q(x)=x^2−2x−1, find the absolute maximum value over the interval [−2,5].

14

Owners of a boat rental company that charges customers between $125 and $325 per day have determined that the number of boats rented per day n can be modeled by the linear function n(p)=1300−4p, where p is the daily rental charge. How much should the company charge each customer per day to maximize revenue? Do not include units or a dollar sign in your answer.

162.5

What can we say about absolute extrema of the function f(x) = e^(sin(x^2)), defined over the entire real line ℝ?

This function attains both its absolute maximum and its absolute minimum, and they are attained at infinitely many critical points.

Find the values of a and b that make the following piecewise function differentiable everywhere. f(x)={−3x2+ax+b −12x+7 for x≤1 for x>1

a=-6 b=4

Find dy/dx, where y is defined as a function of x implicitly by the equation below. y^3+2x^2y^5=1

dy/dx= -(4xy^5)/(10x^2y^4-3y^2

determine limx approaches infinity of (1+7/x)^2x

e^14

The function below is continuous at which of the following values? f(x)=2x−3 3x−1 −2x2+2x if x≤−3 if −3<x≤−1 if −1<x

f(x) is continuous at -1

Determine the infinite limit. limx→0 x+1/x^4(x+5)

infinity

Compute limx→0 ((6^x−2^x)//sinx−x^2)

ln 3

Let a>0 & b>0 be two positive numbers and consider the function f(x) = x^a+x−b. Find the positive value of x where f(x) achieves its minimum value.

x = (b/a)^1/a+b

Find the x-coordinates of all local minima given the following function. f(x)=x^6+3x^5+2

x= -5/2

Use implicit differentiation to find the equation of the tangent line to the function defined implicitly by the equation below at the point (−2,2). x^5−x^3y^2=0 Give your answer in the form y=mx+b.

y=-x

Find the equation of the tangent line to the graph of the function f(x)=4x^(3/2)+1 at x=4.

y=12x-15