Calc quiz 1

integrate x/(sqrt(1 - 9x ^ 2)) dx = (A) - 1/9 * sqrt(1 - 9x ^ 2) + C (B) - 1/18 * ln(sqrt(1 - 9x ^ 2)) + C (C) 1/3 * arcsin(3x) + C (D) x/3 * arcsin(3x) + C

(A) - 1/9 * sqrt(1 - 9x ^ 2) + C

Shown above is a slope field for which of the following differential equations? (A) (dy)/(dx) = (y - 2)/2 (B) (dy)/(dx) = (y ^ 2 - 4)/4 (C) (dy)/(dx) = (x - 2)/2 (D) (dy)/(dx) = (x ^ 2 - 4)/4

(B) (dy)/(dx) = (y ^ 2 - 4)/4

The graph of the function fis shown in the figure above. The value of lim f(x) is (A)-2 (B)-1 (C) 2 (D) nonexistent

(C) 2

The radius of a right circular cylinder is increasing at a rate of 2 units per second. The height of the cylinder is decreasing at a rate of 5 units per second. Which of the following expressions gives the rate at which the volume of the cylinder is changing with respect to time in terms of the radius r and height h of the cylinder? (The volume V of a cylinder with radius r and height h is V=pir^ 2 h.) (A)-20pir (B)-2pirh (C) 4pi*rh - 5pi*r (D) 4pi*rh + 5pi*r

(C) 4pi*rh - 5pi*r

The figure above shows the graph of the continuous function g on the interval [0, 8]. Let h be the function defined by h(x) = integrate g(t) dt from 3 to x .On what intervals is h increasing? (A) [2, 5)only (B) [1,7] (C) [0, 1] and [3,7] (D) [1, 3] and [7,8]

(C) [0, 1] and [3,7]

Which of the following is equivalent to the definite integral integrate sqrt(x) dx from 2 to 6 * 3 (A) lim n -> ∞ sum k=1 to n 4/n * sqrt((4k)/n) (B) lim n -> ∞ sum k=1 ^ n 6 n sqrt 6k n (C) lim n -> ∞ sum k=1 to n 4/n * sqrt(2 + (4k)/n) (D) lim n -> ∞ sum k=1 to n 6/n * sqrt(2 + (6k)/n)

(C) lim n -> ∞ sum k=1 to n 4/n * sqrt(2 + (4k)/n)

The velocity of a particle moving along a straight line is given by v(t) = 1.3fin (0.2t+0.4) for time t20. What is the acceleration of the particle at time t= 1.2? (A) -0.580 (B)-0.548 (C)-0.093 (D) 0.660

(C)-0.093

Honey is poured through a funnel at a rate of r(t) = 4e ^ (- 0.35t) ounces per minute, where t is measured in minutes. How many ounces of honey are poured through the funnel from time t = 0 to time t=3? (A) 0.910 (B) 1.400 (C) 2.600 (D) 7.429

(D) 7.429

Let f be the function with derivative defined by f^ prime (x)=2+(2x-8)sin(x+3) How many points of inflection does the graph offhave on the interval 0<x<9: (A) One (B) Two (C) Three (D)Four

(D) Four

Let f be a twice-differentiable function. Values of f, the derivative of f, at selected values of x are given in the table above. Which of the following statements must be true? (A) f is increasing for - 1 <= x <= 5 (B) The graph of f is concave down for -1<x<5. (C) There exists c, where - 1 < c < 5 such that f^ prime (c)=- 3 2 . (D) There exists c, where - 1 < c < 5 such that f^ prime prime (c)=- 3 2

(D) There exists c, where - 1 < c < 5 such that f^ prime prime (c)=- 3 2

Use the data in the table to approximate R'(5). Show the computations that lead to your answer. Using correct units, explain the meaning of R'(5) in the context of the problem.

-216 At time t=5 hours (12 pm) the rate at which vehicles cross the bridge is decreasing at a rate of approximately 216 vehicles per hour.

Integrate f(x)f'(x)*dx from 0 to 4

-40

On a certain weekend day, the rate at which vehicles cross the bridge is modeled by the function H defined by H(t) = - t ^ 3 - 3t ^ 2 + 288t + 1300for 0≤t≤ 17, where H(t) is measured in vehicles per hour and t is the number of hours since 7:00 A.M. (t = 0) . According to this model, what is the average number of vehicles crossing the bridge per hour on the weekend day for 0 ≤t≤ 12?

2452

Use a midpoint sum with three subintervals of equal length indicated by the data in the table to approximate the value of the integral R(t)*d(t) from 0 to 12. Indicate units of measure.

34,596 vehicles

The table above gives values of the differentiable functions f and g and their derivatives at selected values of x. If h is the function defined by h(x)=f(x)g(x)+2g(x) then h^1(1)= A) 32 B) 30 C) -6 D) -16

A) 32

If x^3-2xy+3y^2=7, then dy/dx= A) (3x^2+4y)/2x B) (3x^2-2y)/(2x-6y) C) 3x^2/2x-6y D) 3x^2/2-6y

B) (3x^2-2y)/(2x-6y)

Let f be the function defined above. At what values of x, if any, is f not differentiable? F(x) = 2/x, x<-1 x^2-3, -1<=x<=2 4x-3, x>2 A) x=-1 only B) x=2 only C) x=-1 and x=-2 D) f is differentiable for all values of x

B) x=2 only

Lim x—>0 (1-cos^2(2x))/(2x)^2 A) 0 B) 1/4 C) 1/2 D) 1

D) 1

For 12 <t<17, L(t), the local linear approximation to the function H given in part (c) at t = 12 is a better model for the rate at which vehicles cross the bridge on the weekend day. Use L(t) to find the time t, for 12 < t <17, at which the rate of vehicles crossing the bridge is 2000 vehicles per hour. Show the work that leads to your answer.

L(t)=H(12)-H'(12)(t-12) H(12)=2596, H'(12)=-216 L(t)=2000 t=14.759

Find the absolute minimum value of f on the interval [0, 4]. Justify your answer.

On the interval [0, 4], the absolute minimum value of fis f(4) = 1 .

On what open intervals contained in (0,4) is the graph of f both decreasing and concave down? Give a reason for your answer.

The graph of f is decreasing and concave down on the intervals (1, 1.6) and (3, 3.5) because f'is negative and decreasing on these intervals.

Find g'(2). Show the work that leads to your answer.

g'(x)=3x²f(x) + x³f'(x) g'(2)=3-2²f(2)+2³ f'(2) = 5 + 8 * 0 = 60