Practice Midterm Exam #1 Math 101 – Single Variable Calculus ...

Practice Midterm Exam #1 Math 101 – Single Variable Calculus Summer 2007

Instructions: This is a 2 hour exam. You may not consult any notes or books during the exam, and no calculators are allowed. 1. Find the largest possible domain for each function. Express this domain as an interval or a union of intervals. √ (a) f (x) = 1 − x2 p (b) g(x) = (x − 2)(x + 1) (c) h(x) =

1 (x−2)(x+1)

2. Write a linear equation for each line described. Give your equation in slope-intercept form. (a) The line L with slope m =

1 2

and y-intercept y = −1.

(b) The line L with slope m = −1 passing through (2, 2). (b) The line L through (1, 4) and (3, −3). 3. For each function, calculate the derivative using the limit definition of the derivative. (a) f (x) = −2x + 1 (b) g(x) = (c) h(x) =

1 x

√

for x 6= 0 2x + 1 for x ≥ − 21 .

4. Calculate each given limit. Justify each step with the appropriate limit rule. (a) limx→1

x2 +1 x3 +3x+2 3

(b) limx→0 cos (x + 3x) √ (c) limx→2 x3 + x − 5

1

5. For each equation, write an equation for the tangent line at the point where x = 1. Write your equation in slope-intercept form. (a) y = x2 + x √ (b) y = 5x2 − x 3

1

(c) y = x 2 + 2x 3 6. For each function, state where it is differentiable and calculate the derivative. Justify each step using the appropriate derivative law or limit law. √ (a) f (x) = 4x6 − 3x5 + x (b) g(x) =

5x3 +6x−1 x−1 2

(c) h(x) = x 5 1/3

(d) j(x) = (x3 − 2x)

7. Let f (x) and g(x) be differentiable at x. (a) Show that, for any h 6= 0 where f (x + h) and g(x + h) are defined, f (x + h) − f (x) g(x + h) − g(x) f (x + h)g(x + h) − f (x)g(x) = g(x+h) +f (x) . h h h (b) Evaluate the limit   g(x + h) − g(x) f (x + h) − f (x) + f (x) lim g(x + h) . h→0 h h Write your answer in terms of f (x), g(x), f 0 (x), and g 0 (x). (c) State the product rule for derivatives and use (a) and (b) to give a proof of it. 8. Find the maximum and minimum of each function on the given interval. (a) f (x) = (b) g(x) =

1 4−x2

√

on the interval [−1, 1].

5x2 − 8x + 4 on the interval [0, 1].

2