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Spring '07 MAT 250 Exam 1: Calculus Limits and Derivatives, Exams of Analytical Geometry and Calculus

The spring 2007 exam 1 for the mat 250 calculus course, focusing on limits and derivatives. Students are required to use the graph of a function, algebraic methods, and the intermediate value theorem to find limits, evaluate continuity, and determine instantaneous velocities. The document also includes a bonus question on exponential expressions.

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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Spring ’07/MAT 250/Exam 1 Name: Show all your work.
1. (7pts) Use the graph of the function to answer the following. Justify your answer if a
limit does not exist.
lim
x3
f(x) =
lim
x3+f(x) =
lim
x0f(x) =
lim
x→−2f(x) =
Is fcontinuous at x=2 and why (not)?
2. (6pts) Find the following limits algebraically.
a) lim
x7
x24x21
x7=
b) lim
x3
5
(x3)2=
pf3
pf4

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Spring ’07/MAT 250/Exam 1 Name: Show all your work.

  1. (7pts) Use the graph of the function to answer the following. Justify your answer if a limit does not exist.

lim x→ 3 −^

f (x) =

lim x→ 3 +^

f (x) =

lim x→ 0 f (x) =

lim x→− 2

f (x) =

Is f continuous at x = −2 and why (not)?

  1. (6pts) Find the following limits algebraically.

a) lim x→ 7

x^2 − 4 x − 21 x − 7

b) lim x→ 3

(x − 3)^2

  1. (9pts) This problem is about the limit lim x→ 4

3 x − 12 √ 8 x + 1 −

a) Use your calculator to estimate the limit with three accurate decimal places. Show the table of values. b) Find the limit algebraically and compare your answer to a).

  1. (5pts) Find lim x→ 0 (x^4 + x^2 )

2 + sin (^1) x. Use the theorem that rhymes with what an allergy

sufferer might do.

  1. (5pts) Is the function f (x) continuous at x = 2? Explain.

f (x) =

3 x − 2 , if x ≤ 2 12 − 4 x, if 2 < x.

  1. (5pts) The graph of f (x) is given. Es- timate the numbers below and draw the graph of f ′(x) under the graph of f (x).

f ′(a) =

f ′(b) =

f ′(c) =

Bonus. (5pts) Algebraically find the limit of the exponential expression

lim x→∞

x^3 +x+ x−x^2 =