MA 16010 final

Find the largest open interval where g(t)g(t) is increasing. g(t)=−1/3t^3+3/2t^2

(0,3)

Find the largest open interval where h(t)h(t) is decreasing. h(t)=t^3−15/2t^2

(0,5)

Find the largest open interval(s) where f(x)=4x^5−5x^4 is concave upward

(3/4,∞)

Given y=x^2sin(x), find y′(π/3).

(√3π/3)+(π^2/18)

Find the x value at which g(x)=xe^x has a horizontal tangent line.

-1

Find the derivative of y=√r^2-10x^2, where r is a constant.

-10x/√r^2-10x^2

The position of an object moving on a straight line is given by s(t)=48−3t−2t^2−6t^3, where tt is in minutes and s(t) is in meters. What is the acceleration when t=3 minutes?

-112m/min^2

Find the critical numbers of y=x^2e^x

-2 and 0

Given that x and y are both differentiable functions of t and xy=6 find dx/dt when x=1/2 and dy/dt=72.

-3

A ball is thrown straight up from the top of a 64-foot building with an initial velocity of 32 feet per second. Use the position function below for free-falling objects and find its velocity after 22 seconds.s(t)=−16t^2+32t+64

-32ft/sec

A 10-ft ladder, whose base is sitting on level ground, is leaning at an angle against a vertical wall when its base starts to slide away from the vertical wall. When the base of the ladder is 6 ft away from the bottom of the vertical wall, the base is sliding away at a rate of 4 ft/sec. At what rate is the vertical distance from the top of the ladder to the ground changing at this moment?

-3ft/sec

Find the slope of the line tangent to xy^2=x^2+y^2 at the point (2,−2).

0

Find the limit numerically: limx→0f(x)=limx→0 e^2x − 1/2x

1

Given that y^2x−x^2=yln(x)+3,use implicit differentiation to find dy/dx at (1,−2).

1

Find the slope of the tangent line to the graph of y=xcotx at x=π/4.

1-π/2

Given h(x)=(x−1)/(8√x+2), find h′(1).

1/10

Given f(x)=ln^3(√3+3x/3-x), find f′(1).

1/3

Given h(t)=3t−1/t√−2 , find h′(16).

1/32

limx→3. (x−2)^3/x+3

1/6

Given y=xlnx, find y″(e).

1/e

What is the slope of the tangent line to the graph of y=ln(2x^3+5x) at x=1?

11/7

The position of a particle moving in a straight line is given by s(t) = 7/3 t^3 − 7t^2 − t + 16 where t is in seconds and s(t) is in feet. What is the particle's position when its velocity is 20 ft/s?

13 ft

The population of a herd of cattle over time (in years) is given byp(t)=70(4+0.1t+0.01t^2). What is the growth rate (in cattle per year) when t=5 years?

14

Find the derivative of f(x)=(x^2−3x)(5x+2)

15x^2-26x-6

An observer stands 400 feet away from the point where a hot air balloon is launched. If the balloon ascends vertically at a (constant) rate of 30 feet per second, how fast is the balloon moving away from the observer 10 seconds after it is launched?

18ft/sec

Find f′(4), given f(x) = x^2 − 1/√ x^3

19/64

Find the limit:limx-1 -4x+4/x^2-4x+3

2

If h(t)=sin(3t)+cos(3t), find h^(3)(t)h.

27sin(3t)−27cos(3t)

A toy rocket is launched from a platform on earth and flies straight up into the air. Its height during the first 10 seconds after launching is given by:s(t)=t^3+3t^2+4t+100. where s is measured in centimeters, and t is in seconds. Find the velocity when the acceleration is 18 cm/s^2.

28cm/s

Given f(x)=x^2/sinx. Find f′(x).

2xsinx−x^2cosx/sin^2x

Given f(x)=3secxtanx. Find f′(x).

3 sec x tan^2 x + 3 sec^3 x

solve limx→−1 x^2 − x − 2/ x^2 − 1 .

3/2

Given f(x)=sin^3(2x), find f′(π/12)

3√3/4

Find lim(x→1-)f(x)if it exists.

4

Find the xx-coordinate of the absolute maximum of g(x)=x^3−3x^2+12 on the closed interval [−2,4].

4

limx→1 x^2+2x−3/x^2+x−2.

4/3

Water flows into a right cylindrical shaped swimming pool with a circular base at a rate of 4 m^3/min. The radius of the base is 3 m. How fast is the water level rising inside the swimming pool?

4/9π m/min

Given f(x)=e^5xln(7x+e) Find f′(0).

5+7/e

A spherical balloon is inflated with gas at a rate of 5 cubic centimeters per minute. How fast is the radius of the balloon changing at the instant when the radius is 4 centimeters? The volume V of a sphere with a radius r is V=4/3πr^3

5/64π centimeters per minute

The derivative of a function is found by computing f ′ (x) = lim h→0 p (5^(√x + h)^2 + 1) − (^5√x^2 + 1)/h

5^√x^2+1

All edges of a cube are expanding at a rate of 2 centimeters per second. How fast is the surface area changing when each edge is 3 centimeters?

72cm^2/sec

I. f(x) is discontinuous at x=−3and x=0. II. lim(x→0)f(x)=−5 III. lim(x→2)f(x) does not exist.

I and II only

choose the correct statement regarding the yy values of the absolute maximum and the absolute minimum of f(x)=x^3−3x+10 on the interval of [0,3].

The y values of the absolute maximum and the absolute minimum are 28 and 8 respectively.

Find dy/dx by implicit differentiation. ln(xy)+2x=e^y

dy/dx=−2xy−y/x−xye^y

Given the derivative, f′(x)=(x+11)^2(x−22), choose the correct statement regarding f(x).

f(x) is increasing on (22,∞).

Given f(x)=e^x(3x^2-x+1). find f′(x).

f(x)=e^x(3x^2+5).

Which of the following functions has a hole at x=4?

f(x)=x^2−16/x−4

The derivative of f(x)=3e^x+cos(x)−2x^3 is:

f′(x)=3e^x−sin(x)−6x^2

Which of the following is TRUE regarding f(x)=x+2/x^2+7x+10

lim(x→-5)f(x)does not exist and f(x)f(x) has a vertical asymptote at x=−5

Which of the following is FALSE regarding f(x) shown in the graph below?

limx--3 f(x)=-2

Which of the following limits does NOT equal to (+∞)?

limx-1- 3x/x^2-1

Given f(x)=1/x+1 , and g(x)=x-1/x^2-1 , which of the following statements is false?

limx-1g(x)does not exist

Which of following does NOT equal to positive infinity (+∞)?

limx-2+ x+8/2-x

Which of the following is TRUE regarding f(x)=x+4/x^2+x-12

limx→3 f(x) does not exist and f(x) has a vertical asymptote at x=3

Given y=tanx(secx+1), find y′.

sec^3x+sec^2x+secxtan^2x

The position of a particle moving on a straight line is given by s(t)=3t^2−12t+9, where t is the time in minutes and s is the position in meters. At what time is the velocity zero?

t=2mins

Find the xx-coordinate of the inflection point of y=e^2x−8x^2

x=1/2ln4

Find the xx value at which the function f(x)=x^3−9x^2−120x+3 has a relative minimum.

x=10

given f(x)=x^2-9/x and f'(x)=x^2+9/x^2. Find the equation of the tangent line to the graph of f(x) at x=−1.

y=10x+18

Find the equation of the tangent line to the graph ofg(x)=x^2+32sqrtx/8 at x=4

y=2x+2

Given f(x)=6sinx. Find the equation of the tangent line to the graph of f(x) at x=π/3

y=3x−π+3√3

Use implicit differentiation to find the equation of the tangent line to the graph at (−2,2). x^2+xy=4−y^2

y=x+4

Given y=csc^2x, find y′.

−2csc^2xcotx

The position of a particle moving on a straight line is s(t) = 2 cos(t) − 1/2 t. Find the velocity function v(t)

−2sin(t)-1/2

Find f′(x) given that f(x)=tan(cos(3x).

−3sin(3x)sec^2(cos(3x))