Midterm 1 Practice Problems

Make sure to spend time studying, too!


Topics Covered



Area and Average Height

1) Find the definite integral.

∫125x4 dx\int_1^2 5x^{4} \ dx

Enter your answer here:



2) Find the definite integral.

∫01x19 dx\int_0^1 x^{19} \ dx

You will need to enter your answer as a decimal, but it should be an easy one to find. [Hint: I'm thinking of a coin. 20 of them make a dollar. What coin am I thinking of?]



3) Find the signed area under the graph and the average height.

∫1432x dx\int_1^4 \frac{3}{2\sqrt{x}} \ dx

Enter the signed area first, then the average height.





Riemann Sums

Highly recommended: Draw a picture for each problem.

1) Calculate the areas of the rectangles for the Right-Hand-Sum of the definite integral below.

∫04x2 dx\int_0^4 x^2 \ dx

n=4.n=4.



2) Calculate the areas of the rectangles for the Right-Hand-Sum of the definite integral below.

∫04x2 dx\int_0^4 x^2 \ dx

n=2.n=2.



3) Calculate the areas of the rectangles for the Left-Hand-Sum of the definite integral below.

∫01x2 dx\int_0^1 x^2 \ dx

n=2.n=2. For areas that aren't whole numbers, enter them as decimals.



4) Calculate the areas of the trapezoids for the Trapezoid-Sum of the definite integral below.

∫0105−x dx\int_0^{10} 5-x \ dx

n=5.n=5.





Derivatives

1) Find the derivative of the following function:

f(x)=x3+2xf(x) = x^3 + 2x


2) Find the derivative of the following function:

f(x)=e5x−xf(x) = e^{5x} - \sqrt{x}

Formatting note: If you need to write write x\sqrt{x}, you can enter x^(1/2). If you need to write 1x\frac{1}{\sqrt{x}}, you can enter either x^(-1/2) or 1/x^(1/2).





Indefinite Integrals

1) Find the indefinite integral.

∫3x2+2 dx\int 3x^2 + 2 \ dx


2) Find the indefinite integral.

∫2x−x5 dx\int 2^x - x^5 \ dx




Word Problems

1a) Find the correct numerical answer for the word problem below.

A valve in a full of 180180 gallon water tank is slowly opening. Water flows out of the tank through the valve. The flow rate in gallons per hour is given by the function f(t)=150t2f(t)=150t^2 where tt is in minutes. How much water flows out of the tank in the first 22 minutes?

[hint: ∫ab( rate per minute ) dt] \left[ \text{\it hint: }\int_a^b \left( \text{ rate per minute } \right) \ dt \right]

Enter your answer accurate to two decimal places.



1b)

How many minutes does it take for the tank to empty completely?

[hint: solve ∫0T150t260 dt = 180] \left[ \text{\it hint: solve }\int_0^T \frac{150t^2}{60} \ dt \ = \ 180 \right]


2)

What is the average of 4x3+2x4x^3+2x for 1≤x≤31\le x\le 3?