Exam 6: Constructing Antiderivatives

The function f(t)is graphed below and we define $F(x)=\int_{0}^{x} f(t) d t$ . Is $F(x)$ concave down for x =1/2?

The rate of change in concentration of a certain medication in a person's body, H'(t), in micrograms per milliliter per minute, is -1 for the first 2 minutes.Then it increases at a constant rate for 2 minutes, reaching 1 at t = 4.Then it remains constant for 1 minute.Sketch H(t), assuming H(0)= 9.

Find the general antiderivative of $P(x)=\frac{1}{2 x}$ .

A factory is dumping pollutants into a lake continuously at the rate of $\frac{t^{2 / 3}}{65}$ tons per week, where t is the time in weeks since the factory commenced operations. Assume that natural processes can remove up to 0.138 tons of pollutant per week from the lake and that there was no pollution in the lake when the factory commenced operations one year ago.How many tons of pollutant have now accumulated in the lake? (Note: The amount of pollutant being dumped into the lake is never negative.)Round to 2 decimal places.

A boulder is dropped from a cliff.A second boulder is dropped from a cliff that is half as high.How does the speed of the second boulder upon impact compare with that of the first?

Evaluate $\int \frac{6 e^{-2 w}}{1-5 e^{-2 w}} d w$ .

Evaluate $\int\left(x^{2}+e^{3 x}\right)\left(x^{3}+e^{3 x}\right)^{2 / 3} d x$ .Some of the coefficients may not be reduced.

A particle is traveling along a straight line.(Positive velocities indicate movement to the right.)What is the total distance traveled (both right and left)when t = 5?

Given the values of the derivative f '(x)in the table and that f (0)= 60, estimate f (4).(Average left- and right-hand sums). x 0 2 4 6 (x) 3 15 27 39

Suppose the rate at which ice in a skating pond is melting is given by $\frac{d V}{d t}=8 t+1$ , where V is the volume of the ice in cubic feet, and t is the time in minutes.Use a definite integral to find how many cubic feet of ice have melted in the first 2 minutes.

A particle is traveling along a straight line.(Positive velocities indicate movement to the right.)Indicate on the graph when the particle has traveled 3 units to the which.

Find an antiderivative of $\frac{5}{\left(a^{2}+x^{2}\right)^{3 / 2}}$ .

A car is going 85 feet per second and the driver puts on the brakes, bringing the car to a stop in 5 seconds.Assume the deceleration of the car is constant while the brakes are on.Suppose a second car is traveling the same speed and the brakes are twice as strong (can stop the car twice as fast)as those in the first car.How far does the second car travel before it stops?

Find an antiderivative of $\sqrt{7 x}-\frac{6}{x \sqrt{x}}$ .

A young girl who aspires to be a rocket scientist launches a model rocket from the ground at time t = 0.The rocket travels straight up in the air, and the following graph shows the upward velocity of the rocket as a function of time: Let v(t)be the function that gives the velocity of the rocket at time t.From the graph of the rocket's velocity, what does the sign of $\int_{0}^{f_{3}} v(t) d t$ mean physically?

Is the antiderivative for $j(x)=a e^{x}+a \cos x+ a x^{7}$ , where a is a constant, given by $a e^{x}+a \sin x+\frac{a x^{8}}{8}+c$ ?

Assuming the 440 feet is accurate and you neglect air resistance, determine the accuracy of the following paragraph: MY JOURNEY BENEATH THE EARTH Condensed from "A Wolverine is Eating My Leg" Tim Cahill: I am in Ellison's Cave, about to rappel down Incredible Pit, the second-deepest cave pit in the continental United States.The drop is 440 feet, about what you'd experience from the top of a 40-story building.If you took the shaft in a free fall, you'd accelerate to more than 100 miles an hour and then--about five seconds into the experience--you'd decelerate to zero.And die.

Find a function F such that $F^{\prime}(x)=x^{5}-\frac{4}{\sqrt{x}}$ .

Use the Fundamental Theorem of Calculus to evaluate $\int_{-2}^{2}\left(9 x^{2}-6 x+5\right) d x$ .

Evaluate $\frac{d}{d x} \int_{2}^{x} \ln \left(t^{3}\right) d t$ .