math test from calculus 1 (2)

Name: Exam 2 - Leighton - MAT 136 - Spring 2020 /100 pts

Read the following completely and carefully. You cannot use electronic devices or notes during this exam. Clearly justify your answers directly on this exam (the last page is 80% blank for this reason); unsupported or illegible answers will not be considered. If you do not know the derivative of a function f, write f′ wherever it is needed. Raise your hand if you have any questions. Good luck!

1 (17 pts). Differentiate the following functions.

(a) a(x) = e2 + xe + ex

(b) b(r) = r3

cos(r)

(c) c(t) = √ t4 + 1

(d) d(w) = 1

w + 5 tan(w)

(E) E(h) = arctan(h) − sec(h) + sin(h)

2 (9 pts). The position, s (feet), of a ball above the ground, t seconds after being thrown, can be modeled by

s(t) = 56 + 62t− 16t2 for 0 ≤ t ≤ 2.608.

One second after being thrown, what is the ball’s (a) position, (b) velocity, and (c) acceleration?

3 (20 pts). Consider f(x) = 3 − 2x2 and the graph of g below. Suppose p(x) = f(x)g(x), q(x) = f(x)/g(x), and c(x) = f(g(x)). Compute the following or explain why they don’t exist.

(a) p′(3)

(b) p′(2)

(c) c′(1.5)

4 (2 pts). It’s known that d

dx

( x2 )

= 2x. Compute the following derivatives if y depends on x but z doesn’t.

(a) d

dx

( y2 )

(b) d

dx

( z2 )

5 (15 pts). Determine equations of all lines tangent to the graph of f(t) = t3/3 + 2t2 + 4t that have slopes of 1.

6 (8 pts). Differentiate f(x) = (3x)ln(x) for x > 0.

7 (5 pts). Determine an equation for dy

dx if y2 + x3 + 2xy = 2 + x and (x, y) 6= (2,−2).

8 (7 pts). Prove that csc′(x) = −csc(x) cot(x).

9 (8 pts). Boyle’s law says that the product of a gas’s pressure, P (atm), and volume, V (m3), is constant. Determine the rate at which the volume of a gas changes when its pressure is 20, its volume is 40, and its pressure is decreasing at a rate of 3 atm per minute.

10 (9 pts). Below is the graph of a function. On the same figure, draw the graph of its derivative. Your graph doesn’t have to be perfect, but it should be positive, zero, or negative wherever appropriate, and it should have the correct values on intervals where they can be straightforwardly computed.