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Math 151 Exam 1: Sample Problems and Solutions, Exams of Calculus

Illinois Institute of Technology (IIT)Calculus

Sample test problems and solutions for a college-level mathematics course, covering topics such as limits, derivatives, and equations. Students can use these problems to prepare for exams, quizzes, or assignments. Problems on functions, calculus, and algebra.

Typology: Exams

Pre 2010

Uploaded on 08/19/2009

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Sample test problems for Math 151-Exam 1

1. Given the function

f(x) = ½4−xfor x≤4

√x−4 for x > 4.

a. Sketch the graph of f(x) then answer the following.

b. lim

x→4f(x)

c. lim

x→∞f(x)

d. Is f(x) continuous? Please explain.

e. Explain why f0(4) does or does not exist?

2. A firework is shot up into the air. The height, S, in feet from the ground is

calculated by the formula: s(t) = −6t2+ 150t+ 1, where tis time in seconds.

a. Use the (limit) definition of derivative to determine S0(a)

b. If a= 6, determine S0(6) .Interpret the meaning of S0(6) and state the

units

c. Find the lim

x→9.38S(t), then explain the meaning in terms of the problem

situation.

3. A lake is stocked with fish and the population of fish that is predicted after

tyears from stocking can be estimated by equation:

f(t) = 5000

1 + 4e−0.2t.

a. What is the maximum number of fish that can be sustained in the lake?

b. What is the domain of f(x) and its range?

4. Using the (limit) definition of derivative, no shortcuts, show that

d

dx (x2−3x+ 1) = 2x−3.

5. Evaluate the following limits:

a. lim

x→1

x2+2x−3

x−1

b. lim

h→0

(3+h)2−9

h

c. lim

x→−∞ ¡√x2−2x+x¢

d. lim

x→−∞e−x2

e. lim

x→0

√5−x−√5

x(x−5)

f. lim

x→∞

√1+4x2

4+x

h. lim

x→3−

|x−3|

2(3−x)

6. If f(x) = x3−x2+x, show that there is a number csuch that f(c) = 10.

7. If f(x) = 3x2−5x, find f0(2) and use it to find to find the equation of the

tangent line to the parabola y= 3x2−5xat the point a= 2.

1

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Sample test problems for Math 151-Exam 1

Given the function

f (x) =

(^4) √ − x for x ≤ 4 x − 4 for x > 4. a. Sketch the graph of f (x) then answer the following. b. lim x→ 4 f (x) c. lim x→∞ f (x) d. Is f (x) continuous? Please explain. e. Explain why f ′(4) does or does not exist?

A firework is shot up into the air. The height, S, in feet from the ground is calculated by the formula: s(t) = − 6 t^2 + 150t + 1, where t is time in seconds. a. Use the (limit) definition of derivative to determine S′^ (a) b. If a = 6, determine S′^ (6). Interpret the meaning of S′^ (6) and state the units c. Find the lim x→ 9. 38 S (t), then explain the meaning in terms of the problem

situation.

A lake is stocked with fish and the population of fish that is predicted after t years from stocking can be estimated by equation:

f (t) =

1 + 4e−^0.^2 t^

a. What is the maximum number of fish that can be sustained in the lake? b. What is the domain of f (x) and its range?

Using the (limit) definition of derivative, no shortcuts, show that

d dx

(x^2 − 3 x + 1) = 2x − 3.

Evaluate the following limits: a. lim x→ 1

x^2 +2x− 3 x− 1 b. lim h→ 0

(3+h)^2 − 9 h c. lim x→−∞

x^2 − 2 x + x

d. lim x→−∞ e−x 2

e. lim x→ 0

√ 5 −x−√ 5 x(x−5) f. lim x→∞

√1+4x 2 4+x h. lim x→ 3 −

|x− 3 | 2(3−x)

If f (x) = x^3 − x^2 + x, show that there is a number c such that f (c) = 10.
If f (x) = 3x^2 − 5 x, find f ′^ (2) and use it to find to find the equation of the tangent line to the parabola y = 3x^2 − 5 x at the point a = 2.

Find the derivatives, dydx , of the following functions: a. h (t) = (^) 1+9tt 2 b. f (x) = 3x^5 + 3

x + (^) x^2 − (^) x^54 + 10 c. m(x) =

x(x^3 −

x + 3x − 10) d. K (x) = (^) 1+tan^1 x e. w(x) = (sin x +

x) cos x f. S(x) = 8 −^3 x

13 5 √x 7 h. g (x) = (^) (1+seccot^ x x) i. p (x) = x^3 sin x cos x

Determine the equation of the tangent line to the curve y =

x^3 + x^3 at the point (4, 72).

At what point on the curve y = x

x is the tangent line parallel to the line 3 x − y + 6 = 0.

a. For what values of x is the function f (x) = |x^2 − 9 | differentiable? b. Sketch the graphs of f and f ′.

Express the function

H(x) =

√ (^3) x + √x

as a composition of three functions, i.e. H (x) = (f ◦ g ◦ h)(x).

Sketch the graph of the following functions: a. y = 3 − (^12)

4 − 3 x^2 b. y = 2 + 3 cos(x − 1)

Find the vertical and horizontal asymptotes of the function y = (^) x (^2) −xx− 2
Find all values of x in the interval [0, 2 π] that satisfy the equation

2 cos x + sin 2x = 0.

Identify the type of the curve and sketch the graph for

2 x^2 + 2y^2 − x + y = 1.

Find the equation of the line through (− 1 , −2) and perpendicular to the line 2x + 5y + 8 = 0.
Solve the equation

∣ (^2) xx+1−^1

∣ = 3 for x.

Math 151 Exam 1: Sample Problems and Solutions, Exams of Calculus

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