NAME:

Math 180, Calculus I Hour Exam Two 10:00 am Lecture November 3, 1995 1. The function f(x) is increasing. Some of its values are given in the table:

\begin{matrix}x & 0 & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 & 3.5 \\ f(x) & 1.0 & 1.2 & 1.6 & 2.0 & 2.4 & 2.6 & 3.2 & 3.4 \end{matrix}

(i) Compute the left and right Riemann sums with two subdivisions (n = 2) for the

integral

∫_{2}^{3} f(x) dx.

(ii) How do these sums compare with the value of the integral?

2. Find the derivatives of the following functions. Please do not simplify your answers.

(i) x^{1642} − 2x^{5} + x^{1/3} + π
(ii) \frac{x}{x^{3} + 1}
(iii) e^{−x} sin x

3. (i) Find a function whose derivative is 4x + 1.

(ii) Evaluate: ∫_{1}^{b}(4x + 1) dx.

4. The temperature of a pie in a 325^{∘} oven is given by

f(t) = 325 − 255e^{−0.1t}

where t is the time (in minutes) the pie has been in the oven.

(i) Write a formula using an integral for the average temperature of the pie during the first 30 minutes, 0 ≤ t ≤ 30.

(ii) Calculate this average temperature with an error of at most 5^{∘}.

5. For the curve xy + 3y^{2} = 18, (i) Find \frac{dy}{dx} in terms of x and y. (ii) Write the equation of the tangent line at the point (3, 2).

6. Differentiate the following functions and show your work.

(i) ln(x + x^{3})
(ii) e^{cos(\sqrt{x})}

(iii) tan(3x − 2).