math test from calculus 1(1)

MAT 136: Calculus I - Spring 2020

MAT 136: Calculus I - Spring 2020 Exam 1

NAME: Class grade: Test grade /58.5

Instructions: Answer each of the following questions completely. To receive full credit, you must show sufficient work for each of your answers (unless stated otherwise). How you reach your answer is more important than the answer itself.

1. Draw a graph with the following properties. (5 points).

a) lim x→∞

f(x) = −∞

b) lim x→−∞

f(x) = 5

c) f has a jump discontinuity at x = 5

d) lim x→−5

f(x) = ∞

e) f ′(x) = 0 at x = 0.

2. Draw a graph where lim x→2

f(x) exists but f(x) is discontinuous at x = 2. Name the type of discontinuity

that has this property. (4 points).

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MAT 136: Calculus I - Spring 2020

3. Let f be a function such that −x2 < f(x) < x2 for the interval [−10, 10]. Does the squeeze theorem apply when finding limx→1 f(x)? If so find the limit. If not, for what x value is the squeeze theorem viable? (8 points)

4. Let f(x) = 3x2 + 5x. Using the limit definition of derivative prove that f ′(x) = 6x + 5. Only prove this using the limit definition; any other method will not count. (8 points).

5. Let f be the same function as in question 4. Find the tangent line of f(x) at x = 3. (3 points)

6. Let f be the same function as in question 4. Find the average rate of change between x = −1 and x = 2. (3 points)

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MAT 136: Calculus I - Spring 2020

7. Let a function f be described via the graph below. Find the following limits if a limit doesn’t exist write DNE. (1 point each)

a. lim x→−3

f(x).

b. lim x→−6−

f(x)

c. lim x→−6+

f(x)

d. limx→−6 f(x)

e. lim x→∞

f(x)

f. lim x→−∞

f(x)

g. lim x→2+

f(x)

h. lim x→2−

f(x) i. lim x→2

f(x) j. lim x→6

f(x)

8. Find the following limits using any method discussed so far in the class. If the limit does not exist write DNE. (2.5 points each)

a) lim x→0

x3 cos( 5 x )

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MAT 136: Calculus I - Spring 2020

b) lim x→5

x2 − 10x + 25 x2 − 25x + 50

c) lim x→−∞

−7x5 − 35x2 + 15x 4x5 + 3x3 − 20

d) lim x→0

sin(3x) cos(x) − sin(3x) x2

e) lim x→4

x− 4 √ x− √

8 −x

9. Let f be a function. Explain, using any method , as to why f(x) and f(x) + 2 have the same derivative. (5 points)

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