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Math 1441: Practice Test #1 - Derivatives and Limits, Exams of Calculus

Georgia Southern University (GS)Calculus

A practice test for math 1441, focusing on derivatives and limits. It includes matching questions, derivative computations, finding equations of tangents, and evaluating limits. Students are required to show their work and box their answers.

Typology: Exams

Pre 2010

Uploaded on 02/24/2010

koofers-user-tlo
koofers-user-tlo 🇺🇸

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Math 1441 Practice Test #1 Name
F. Ziegler Fall 2005
Show your work to receive credit, and box your final answer in every computation.
1. Match the graph of each function in (a–d) with the graph of its derivative in (1–4).
x
y
y
x
x
y
x
y
(a) (b) (c) (d)
x
y
x
y
x
y
y
x
(1) (2) (3) (4)
2. Compute the derivatives of the following functions:
(a) f(x) = x4−23
√x+1
x4
(b) f(x) = (3x5+ 2x)10
pf3
pf4

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Math 1441F. Ziegler Practice Test #1 Name Fall 2005

Show your work to receive credit, and box your final answer in every computation.

  1. Match the graph of each function in (a–d) with the graph of its derivative in (1–4).

x y y

x x y

x y (a) (b) (c) (d)

x y

x y

x y

y x (1) (2) (3) (4)

  1. Compute the derivatives of the following functions: (a) f (x) = x^4 − 2 √^3 x + x^14

(b) f (x) = (3x^5 + 2x)^10

(c) y = (x^2 + 1) sin(− 3 x^2 − x)

(d) y = cossin xx + 2+ 4

(e) y = tan(sin(x^2 + 1))

  1. Find an equation of the tangent to the curve y = sin x at the point (π, 0).
  2. Compute the derivative of f (x) = 3 + x^1 using only the definition of f ′(x) as a limit. (Show all steps; no credit for using a ready differentiation formula.)

3

3

2

y

1 (^02) ! 2

! 10 1 ! 3

! 3! (^2) x! 1

  1. Pictured below is the graph of f (x) = (^) x^13. (a) Explain exactly what is wrong with the following ar-gument: “We have f (−1) = −1 and f (1) = 1. There- fore, by the intermediate value theorem, the interval[− 1 , 1] must contain a point x such that f (x) = 0.”

(b) Find the limits (or say if they don’t exist): xlim→− 1 f^ (x) xlim→ 0 − f^ (x) xlim→0+ f^ (x)

  1. It is known that 1 + x 6 ex^6 (1 − x)−^1 for every x in the interval (− 1 , 1). (a) Using this fact and the squeeze theorem, find the value of lim x→ 0 ex.

(b) What does the result of (a) say about the continuity of f (x) = ex^ at x = 0?