Practice Problems
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Given the distance function, s(t), where s is in feet and t is in seconds, find the velocity function, v(t), and the
acceleration function, a(t).
1) s(t) = 3t2 + t + 10
Differentiate.
2) g(x) =
x2 + 5
x2 + 6x
Find an expression for dy/dx.
3) y = (u + 6)(u - 6) and u = x2 + 9
Calculate the requested derivative from the given information.
x+4
4) Given f(u) = u3 and g(x) = u =
, find (f † g)'(0).
x-2
Find the derivative.
5) f(x) = e8x ln x
Differentiate.
6) y = ln [ln x]5
For the given function, find the requested relative extrema or extreme value.
7) y = x2 e8x; minimum value on [-2, 0]
Find all relative maxima or minima.
x3
8) f(x) =
6lnx
Find the equation of the line tangent to the graph of the function at the indicated point.
9) y = (x2 + 28)4/5 at x = 2
Write an equation of the tangent line to the graph of y = f(x) at the point on the graph where x has the indicated value.
-4x2 - 4
10) f(x) =
,x=0
4x - 1
Find the indicated tangent line.
11) Find the tangent line to the graph of f(x) = -6e7x at the point (0, -6).
Find the limit, if it exists.
6x3 + 4x2
12) lim
xŒ‹ 7x2 - x
1
Sketch a graph of the function.
7
13) f(x) = x +
x
10
y
8
6
4
2
-10 -8 -6 -4 -2-2
2
4
6
8
x
-4
-6
-8
-10
Find the relative extrema of the function, if they exist.
14) f(x) = x4/3 - x2/3
Solve the problem.
15) A piece of land is shaped like a right triangle. Two people start at the right angle at the same time, and walk at
the same speed along different legs of the triangle while spraying the land. If the area covered is changing at
4 m2 /s, how fast are the people moving when they are 5 m from the right angle? (Round
approximations to two decimal places.)
Differentiate implicitly to find the slope of the curve at the given point.
16) xy3 - x5 y2 = -4; (-1, 2)
2
Answer Key
Testname: PRACTICE1.TST
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
1) v(t) = 6t + 1; a(t) = 6
6x2 - 10x - 30
2) g'(x) =
x2 (x + 6)2
3) 4x(x2 + 9)
4) - 18
e8x (1 + 8x ln x)
5)
x
6)
5
x ln x
7) 0
1
8) e1/3 , e , relative minimum
2
9) y =
8
64
x+
5
5
10) y = 16x + 4
11) y = -42x - 6
12) ‹
13)
10
y
8
6
4
2
-10 -8 -6 -4 -2
-2
2
4
6
8
x
-4
-6
-8
-10
14) (0, 0), ( 2/4, - 1/4), (15) 0.80 m/s
3
16) 2
2/4, - 1/4)
1