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Math 125 Exam Review: Derivatives, Integrals, Limits, and Functions, Exercises of Calculus

Practice problems from the first, second, third, and final exams of a college-level mathematics 125 course. Topics covered include limits, derivatives, integrals, and functions. Students are asked to find limits, derivatives, and integrals of various functions, as well as identify intervals of increase, decrease, concavity, and local extrema. Some problems involve interpreting physical situations and making estimates.

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Math 125 - First Exam

1. (20 pts)

(a) For y=f(x), define the derivative of fat a,f



(a).

(b) What three things does the derivative do?

2. (35 pts) For the function whose graph is pictured below,

1

(a) lim

x



0f(x) =

(b) lim

x



2f(x) =

(c) lim

x



2−

f(x) =

(d) lim

x



2+f(x) =

(e) lim

x



f(x) =

(f) f



(0) =

(g) f

 

(4) =

(h) At what points is fNOT continuous?

(i) At what points is fNOT



tiable?

(j) On what intervals is f

 

(x)>0?

(k) Sketch (on the same picture) the graph of f



.

3. (20 pts) Let f(t) be the temperature of a cup of





in degrees Fahrenheit, t

minutes after it has been poured.

(a) Interpret f(4) = 120 and f



(4) = −5. What are the units of each.

(b) Estimate the temperature of the





after 5 minutes and after 8 minutes.

What can you say about the accuracy of these estimates?

4. (25 pts) Find the following.

(a) lim

x



3

x2+ 9

x+ 3

(b) lim

x



3

x2+ 9

x−3

(c) lim

x



3

x2

−9

x−3

(d) Using the definition, f



(3) for f(x) = x2.

(e) g



and g

 

for ggiven by g(t) = 144 + 96t−16t2.

Math 125 - Second Exam

1. (30 pts) Find the derivative with respect to x

(a) y=x3

−2x+ 1

(b) f(x) = e−x2

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Math 125 - First Exam

(20 pts)

(a) For y = f (x), define the derivative of f at a, f

(a). (b) What three things does the derivative do?

(35 pts) For the function whose graph is pictured below,

1 1

(a) lim x 0 f (x) =

(b) xlim 2 f (x) =

(c) x lim 2 − f (x) =

(d) x lim 2 + f (x) =

(e) xlim f (x) =

(f) f

(g) f

(h) At what points is f NOT continuous?

(i) At what points is f NOT ^ ^ tiable?

(j) On what intervals is f

(x) > 0? (k) Sketch (on the same picture) the graph of f

3. (20 pts) Let f (t) be the temperature of a cup of ^ in degrees Fahrenheit, t

minutes after it has been poured. (a) Interpret f (4) = 120 and f

(4) = −5. What are the units of each.

(b) Estimate the temperature of the after 5 minutes and after 8 minutes.

What can you say about the accuracy of these estimates?

(25 pts) Find the following.

(a) lim x 3 x

x + 3

(b) lim x 3 x

x − 3

(c) lim x 3 x

x − 3

(d) Using the definition, f (3) for f (x) = x^2.

(e) g and g for g given by g(t) = 144 + 96t − 16 t^2.

Math 125 - Second Exam

(30 pts) Find the derivative with respect to x

(a) y = x^3 − 2 x + 1 (b) f (x) = e−x^2

(c) g(x) = tan−^1 x (d) h(x) = ln 3 (e) s = t^2 + 1 (f) z = x ln(x) − x

(20 pts) Find intervals on which F (x) = xe−x^ is increasing, decreasing, concave up, concave down, local maxima and minima, and inflection points.
(15 pts) Find all points on the curve x^2 + 2y^2 = 1 where the slope of the tangent line is 1.
(10 pts) A plane flying with a constant speed of 300 km/h passes over a radar station

at an altitude of 1 km and climbs at an angle of 30. At what rate is the distance

from the plane to the radar station increasing a minute later?

(15 pts) The hyperbolic sine and cosine are defined by sinh x = (ex^ − e−x)/2 and cosh x = (ex^ + e−x)/2 and other hyperbolic “trig” functions are defined analgous to corresponding trigonometric functions. (a) Show that cosh^2 x − sinh^2 x = 1. (b) Show that (^) dxd sinh x = cosh x. Find the derivative of cosh.
(10 pts) If the length of each edge of a cubical box is increased by 1% approximately what is the percent increase in the volume? Math 125 - Third Exam
(40 pts) Evaluate the following:

(a) lim z 0 tan z^ z

(b) tlim t^2 e−^2 t

(c) x^2 − 3 sin x + 3 x dx

(d) lim

0 +

cos

(e)

3 1

(y − 1)(y − 2) dy

2. (15 pts) A rock is dropped from the top of a and hits the valley below in 5

seconds. Assuming air resistance in negligible, determine how high the

is.

(15 pts) An animal population is increasing at a rate of 200 + 50t per year (where t is measured in years). By how much does the animal population increase between the fourth and tenth years?
(15 pts) The velocity of a wave of length L in deep water is

v = K (^) CL + C L

where K and C are known positive constants. What is the length of the wave that gives the minimum velocity?

Math 125 Exam Review: Derivatives, Integrals, Limits, and Functions, Exercises of Calculus

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Partial preview of the text

Download Math 125 Exam Review: Derivatives, Integrals, Limits, and Functions and more Exercises Calculus in PDF only on Docsity!

(a) lim x 0 f (x) =

(b) xlim 2 f (x) =

(c) x lim 2 − f (x) =

(d) x lim 2 + f (x) =

(e) xlim f (x) =

(i) At what points is f NOT ^  ^  tiable?

3. (20 pts) Let f (t) be the temperature of a cup of   ^ in degrees Fahrenheit, t

(b) Estimate the temperature of the   after 5 minutes and after 8 minutes.

(a) lim x 3 x

(b) lim x 3 x

(c) lim x 3 x

(d) Using the definition, f (3) for f (x) = x^2.

(e) g and g for g given by g(t) = 144 + 96t − 16 t^2.

at an altitude of 1 km and climbs at an angle of 30. At what rate is the distance

(a) lim z 0 tan z^ z

(b) tlim  t^2 e−^2 t

(c) x^2 − 3 sin x + 3 x dx

(d) lim

cos 

2. (15 pts) A rock is dropped from the top of a  and hits the valley below in 5

seconds. Assuming air resistance in negligible, determine how high the  

(i) At what points is f NOT ^ ^ tiable?

3. (20 pts) Let f (t) be the temperature of a cup of ^ in degrees Fahrenheit, t

(b) Estimate the temperature of the after 5 minutes and after 8 minutes.

at an altitude of 1 km and climbs at an angle of 30. At what rate is the distance

(b) tlim t^2 e−^2 t

(c) x^2 − 3 sin x + 3 x dx

(d) lim

cos

2. (15 pts) A rock is dropped from the top of a and hits the valley below in 5

seconds. Assuming air resistance in negligible, determine how high the