Calc Exam II

Find the point on the graph of y = √x between (1,1) and (9,3) at which the tangent to the graph has the same slope as the line through (1,1) and (9,3)

(4,2)

The area of the first quadrant region bounded above by the graph of y = e^x^2 and the line x=1 is ....

(e - 1) / 2

The function f is continuous at x = 1. If f(x) = {√(x+3) - √(3x+1) for x not = 1 and k for x = 1} what does k =?

-1/2

Let f(x) = x ln x. The minimum value attained by f is ....

-1/e

The function f is continuous and differentiable on the closed interval [1,5]. The table below gives selected values of f on this interval. Which of the following statements must be true? A) f'(x) > 0 for 1<x<3 B) f"(x) < 0 for 3<x<5 C) The minimum number of f on [1,5] must be -2 D) There exists a number c, 1<c<5 for which f(c) = 0 (1,3) (2,4) (3,5) (4,3) (5,-2)

D

The slope of the tangent line to the curve 2xy + sin y = 2pi at the point where y = pi is .....

-2pi

lim ((cos(x+h) - cos x)) / h as lim h—-> 0 = ....

-sinx

The acceleration of a particle at time t moving along the x-axis is given by: a = 4e^2t. At the instant when t = 0, the particle is at the point x = 2 moving with velocity v = -2. The position of the particle at t = 1/2 is ....

e - 1

pi/3 ∫ pi/4 (sec^2x / tanx) dx = ....

ln √3

The region bounded by the x-axis and the part of the graph of y = sin x between x = 0 and x = pi is separated into two regions by the line x = k. If the area of the region for 0 < x < k is 1/3 the area of the region for k < x < pi, then k = .....

pi/3

A particle starts at time t = 0 and moved along a number line so that it's position, at time r > 0 is given by x(t) = (t-2)^3 (t-6). The particle is moving to the right for ......

t > 5

An equation of the normal to the graph of f(x) = x / (2x-3) at (1, f(1)) = ...

x - 3y = 4

What is the solution to the differential equation dy/dx = y^2, where y(-1) = 1?

y = -1/x for x < 0

Let f(x) = 4x^3 - 3x - 1. An equation of the line tangent to the graph of y = f(x) at x=2 is...?

y = 45x - 65

On which of the following intervals, is the graph of the curve y = x^5 - 5x^4 + 10x + 15 concave up 1. x < 0 2. 0 < x < 3 3. x > 3

3 only

If f(x) = e^(2x) and g(x) = ln x, then the derivative of y = f(g(x)) at x = e is ....

2e

The function f is defined by f(x) = (x-2)^(2/3) + 1. The absolute minimum value of f on the closed interval [1,10] is ....

1

The graph of f', the derivative of function f, is shown on the last page of the packet. The graph of f' has a horizontal tangent at x = 0. Which of the following statements are true about function f? 1. f is increasing on the interval (-2, -1) 2. f has an inflection point at x = 0 3. f is concave up on the interval (-1, 0)

1 and 2

The derivative of √x - 1/(x 3√x) is ...

1/2 x^(-1/2) + 4/3 x^(-7/3)

Integral (arctan x) / (1+x^2) dx = ....

1/2(arctan x)^2 + C

If g(x) = (x - 2) / (x + 2) then g'(2) = ....

1/4

lim (1-cosx) / (2sin^2x) as x —> 0 is ....

1/4

The antiderivative of (x^2 -1)^2 is .....

1/5x^5 - 2/3x^3 + x + C

If 4 ∫ 2 f(x) dx = 6, then 4 ∫ 2 (f(x) + 3) dx = ....

12

2 ∫ 0 (√(x^2 - 4x + 4) dx is ....

2

If the function G is defined for all real numbers by G = 2x ∫ 0 cos(t^2) dt, then G'(√pi) is .....

2

The average value of sec^2 x over the interval 0 < x < pi/4 is ...

4/pi

Propane is pumped into a tank at a constant rate of 6 gallons per minute. Propane leaks out of the tank at the rate of 1/(√(t+1) gallons per minute. At t = 0, the tank contains 20 gallons of propane. How many gallons of propane are in the tank at t = 8 minutes?

64

The slope field for a differential equation dy/dx = f(x,y) is given in the figure. The slope field corresponds to what differential equation? A) dy/dx = x + y B) dy/dx = y^2 C) dy/dx = -y D) dy/dx = e^-x

B