calculus

let f(x)=1/x^2 and g(x)=3x+5. Then, (fog)(x) is

1/(3x+5)^2

For f(x) = sqrt(2 +√x), evaluate f'(4).

1/16

Find the limit as approaches 0 sqrt(1+x)-1/x

1/2

Let y=sqrtu and u=7x-2x^2. Find dy/dx.

1/2(7-4x)(7x-2x^2)^-1/2

Evaluate: limit as z approaches infinity (x^2-1)/(3x^2-2)

1/3

Find the area of the region under the graph of function f(x) = x^2 on the interval [0, 1].

1/3

The second derivative of function f(x)=(x^2+1)^5 is

10(x^2+1)^3(9x^2+1)

It costs an artist 1000 + 5x dollars to produce x signed prints of one of her drawings. The price at which x prints will sell is 400/√x dollars per print. How many prints should she make in order to maximize her profit ?

1600

An open box is to be made from a square sheet of tin measuring 12 inches × 12 inches by cutting out a square of side x inches from each corner of the sheet and folding up the four resulting flaps. To maximize the volume of the box, take x =

2

The total weekly cost on dollars incurred by Herald Media Corp. in producing x DVDs is given by the total cost function C(x) = 2500 + 2.2x, 0 ≤ x ≤ 8000. The marginal cost and the average cost function are...

2.2, 2500/x+2.2

Evaluate S(√x − 2e^x) dx.

2/3x^(3/2) − 2e^x + C

Evaluate: limit as x approaches 3 (3x^2-4)

23

Find dy/dx in terms of x only when x and y are related by the equation ln y=2x-3.

2e^(2x-3)

Let f(x)= x^2-4x. When simplified, the difference quotient f(x+h)-f(x)/h becomes

2x+h-4

Find the limit as x approaches infinity (3x^2+2x+4)/(2x^2-3x+1)

3/2

Let f(x)=x^2+1 and g(x)=1/squareroot x. Then f(g(2)) is

3/2

Given the demand equation 4x+2p−36 = 0 and the supply equation 2x−p+10 = 0, where p is the unit price and x represents the quantity, find the equilibrium quantity and the equilibrium price.

x=2 and p=14

What is the profit in terms of the cost C(x) to produce x units and the unit price p(x) at which x units will sell?

xp(x)-C(x)

Find an equation of the tangent line to the graph of y=e^(−x^2) at the point (1, 1/e)

y = −2/e(x − 1) + 1/e

Find the derivative of function y = x^(ln x). (Hint: use logarithmic differentiation.)

y' = ((2 ln x)/x) x^ln x

Find the derivative of function y = 10^x. (Hint: use logarithmic differentiation.)

y'= 10xln 10

The line tangent to y = x^2−3x through the point (1,-2) has equation

y=-x-1

Find the horizontal asymptotes of function f(x)= (x^2)/(1+4x^2)

y=1/4

An equation for the line tangent to the curve y = x^3 −2x +5 at the point (-2, 1) is

y=10x+21

Find an equation of the tangent line at the point (1, 32) of the graph of y = x(x+1)^5

y=112x-80

An equation for the line tangent to the curve y = (x^2 + x + 1)(x^3 − 2x + 2) at the point (1, 3) is

y=6x-3

Find an equation of the tangent line to the graph of y= x ln x at the point (1,0).

y=x-1

Find an equation of the tangent line to the graph of y= ln(x^2) at the point (2, ln 4).

y=x-2+ln4

Let f(x)= ln(2-x). The domain of f is

(-infinity, 2)

Evaluate S2x(x^2 + 3)^10dx. (Hint: Use substitution)

(1/11)(x^2+3)^11+c

Find dy/dx in terms of x and y when x and y are related by equations x^2y-y^3=2.

(2xy)/(3y^2-x^2)

The velocity of a car (in feet/second) t seconds after starting from rest is given by the function f(t) = 2√t (0 ≤ t ≤ 30). Find the car's position at any time t.

(4/3)t^(3/2)

Find the marginal revenue for the revenue function 50 − p(x 2 + 1) = 0.

(50(1-x^2))/(x^2+1)^2

Let Let f(x) = 1/3x^3−x^2 + x − 6. Find the inflection points, if any.

(x, y) = (1, f(1))

Let f(x) = e^−x^2. Find the inflection points, if any.

(x, y) = (−1√2, f(−1√2)) and (x, y) = ( 1√2, f(1√2))

1. The domain of function f(x) =( x+3)/(2x^2−x−3)

(−∞, −1) ∪ (−1,3/2) ∪ (3/2,+∞)

Find dy/dx in terms of x and y when x and y are related by equation 2x^2+y^2+2x^2y^2+10.

-(x(1+2y^2)/y(1+2x^2)

Find dy/dx in terms of x and y when x and y are related by the equation x^1/3+y^1/3=1

-(y/x)^2/3

Let f(x)=1/x. When simplified, the difference quotient f(x+h)-f(x)/h becomes

-1/x(x+h)

Suppose h=fog. Find h'(0) given that f(0)+6, f'(5)=-2, g(0)=5, and g'(0)=3.

-6

The third derivative of f(x)=1/x is

-6/x^4

The distance s (in feet) covered by a car t seconds after starting from rest is given by s=-t^3+8t^2+2-t. Find the car's acceleration at time t.

-6t+16

The differential of function f(x) = 1000 is

0

Use differentials to estimate the change in √x^2 + 5 when x increases from 2 to 2.123.

0.082

A ball is thrown straightly up into the air so that its height (in feet) after t seconds is given by s(t) = −16t^2 + 64t. The average velocity of the ball over the interval [1, 1.05] is... Assume that the distance function s(t) is given as in Problem 16. The velocity of the ball at time t = 1 is

31.2 ft/sec, 32 ft/sec

Find dy/dx at point (2, sqrt 5) when x and y are related by the equation 2x^2-y^2=3

4/sqrt5

The unit price p and the quantity x demanded are related by the demand equation 50 − p(x 2 + 1) = 0. Find the revenue function R = R(x).

50x/(x^2+1)

Calculate RS8-1(4x^1/3+8/x^2)dx.

52

Find limit as x approaches 3 (x^2-9)/(x-3)

6

Find the area of the region under the graph of y = x^2+1 from x=−1 to x=2.

6

Suppose that F(x)=f(x)^2+1, f(1)=1, and f'(1)=3. find F'(1).

6

Suppose that F(x)=f(x^2+1) and f'(2)=3. Find F'(1).

6

Evaluate: limit as x approaches 5 (x^2-2x-15)/(x-5).

8

Let f(x) = x ln x. Find the inflection points, if any

No inflection points

Find the limit as z approaches negative infinity (x^2+3)/(x+1)

None of the above

Find the second derivative of the function f(x) = e x + ln(x^2) + x ln 3 + 10.

None of the above

The derivative of the function f(x)= ((x-3)/(sqrt x+1)) + (sqrt3x+4)+3

None of the above

Let f(x)= (sqrt x+1)/(x-2). The domain of f is

[-1, 2) and (2, +infinity)

The absolute maximum value and the absolute minimum value of the function f(x) =1/2^x2 − 2√x on [0, 3] are

absolute min. value: -3/2; absolute max. value: 9/2 -2square root of 3

Find the absolute extrema of the function f(t) = (ln t)/t on [1, 2].

absolute min. value: 0; absolute max. value: ln 2/2

Let Let f(x) = 1/3x^3−x^2 + x − 6. Find the intervals where f is concave upward and where it is concave downward.

concave downward on (−∞, 1) and upward on (1,∞)

Let f(x) = x ln x. Determine the intervals of concavity for the function.

concave upward on (0, ∞)

Let f(x) = e^−x^2. Find the intervals where f is concave upward and where it is concave downward.

concave upward on (−∞, −1√2) and on ( 1√2,∞); concave downward on (−1√2,1√2)

Let f(x) = x ln x. Determine the intervals where the function is increasing and where it is decreasing.

decreasing on (0,1/e) and increasing on (1/e,∞)

Evaluate limit as z approaches 1- f(x) for the function f defined as f(x)= {(e^x, for 0<x<1)(3, for x=1)(lnx, for x>1)

e

Let f(x) = {2x-4 if x<0 1 if x>0,

f(x) is discontinuous at x=0

It is known that limit as x approaches 2+ f(x)=3, limit as x approaches 2- f(x) =3, and f(2)=1.

f(x) is discontinuous at x=2, The graph of f(x) is broken at x=2, lim as x approaches 2 f(x) exists, f(x) is defined at x=2

Let f(x) = 1/3x^3−x^2 + x − 6. Determine the intervals where the function is increasing and where it is decreasing.

increasing on (-infinity, 1) and on (1, infinity)

Let f(x) = e^−x^2. Determine the intervals where the function is increasing and where it is decreasing

increasing on (−∞, 0) and decreasing on (0,∞)

Find the absolute maximum value and the absolute minimum value, if any, of the function f(x) = 1/(1+x^2).

no absolute min. value; absolute max. value: 1

Find the absolute extrema of function f(t) = te^−t

no absolute min. value; absolute max. value: 1/e

Let f(x) = e^−x^2. Find the relative extrema of f.

no relative min. value ; relative max. value: 1

Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than 108 inches. Find the dimensions of the cylindrical package of greatest volume that may be sent through the mail. (In the answers, r is the radius and l is the length.)

r x l= 36/pi *36

A rectangular box is to have a square base and a volume of 20ft^3 . If the material for the base costs 30 cents/square, the material for the four sides costs 10 cents/square, and the material for the top costs 20 cents/square, determine the dimensions of the box that can be constructed at minimum cost.

x*x*h=2*2*5

Let f(x)=x/x^2+1 and g(x)=1/x. Then (gof)(x) is

x+1/x

Find the vertical vertical asymptotes of function f(x)= (2+x)/(1-z)^2.

x=1