MTH 263 NVCC Properties of Logarithmic Function Calculus Exam Summary is pretty clear. 10 calculus 1 problems that I need solved with ALL the work shown please. Write on this Test
8/18 – Version A
Northern Virginia Community College – NOVA Online
MTH 263 – Calculus I
Exam 2_A – Form 8-18-A
(Cover Sheet plus 7 pages)
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MTH 263 Form 8-18-A
Directions: Read each question carefully. Show All of your work to earn full credit. All steps
must be shown! Please make sure your work is neatly written and organized.
1. Find the derivative of the following functions:
f ( x) = sin 3 (cos(2 x))
y = ecosh5x
Use the properties of logarithmic function to rewrite y before you find the derivative:
x2 + 3
y = ln
( 2 x + 5 )
Find f (4) ( x) for f ( x) =
2. Use the given graph of the derivative f ′ of a continuous function f over the interval
( 0,9 ) .
To find the following:
(a) On what interval(s) is f increasing?
(b) On what interval(s) is f decreasing?
(c) At what values of x does f have a local maximum or minimum?
3. Use implicit differentiation to find an equation of the tangent line to the graph of
x 2 + 4 xy + y 2 =
13 at the point ( 2,1) .
4. Let f ( x)= sin( x + π ) + cos 2 x
(a) Compute f ′( x), f ′′( x), f (3) ( x), f (4) ( x)
(b) Observe the pattern in part (a) and use it to compute f (9) (0)
5. (a) A plane flying horizontally at an altitude of 3 mi and a speed of 480 mi/h passes
directly over a radar station. Find the rate at which the distance from the plane
to the station is increasing when it is 4 mi away from the station.
(b) A particle moves with position function s (t ) =t 4 − 4t 3 − 20t 2 + 20t , t ≥ 0 .
At what time does the particle have a velocity of 20 m/s ?
Find an equation of the tangent line to the curve at the given point.
, (0, 2)
1 + e− x
(b) True or false. If it is false, explain why it is false.
x + x = 2x +1
7. Answer the following:
f ( x) tan
x and a
i. Find the equation of the line that represents the linear approximation
to the function f at the given value a =
ii. Use the linear approximation of f you found in (i) to estimate the function value
(b) Find the differential dy given y = sec 2 x
Given the function f ( x) =
(a) all critical numbers
x4 + 1
. Find each of the following.
(b) intervals that f is concave up
(c) classify all local extrema
(d) Sketch the graph of f .
9. (a) At 2:00 PM a car’s speedometer reads 30 mi/h. At 2:15 PM it reads 50 mi/h.
Assume the velocity function of the car is differentiable. Use the Mean Value Theorem
to show that at some time between 2:00 and 2:15 the acceleration is exactly
(b) Sketch the graph of a function that satisfies all of the given conditions.
f ′( x) > 0 if x ≠ 2, f ′′( x) > 0 if x < 2, f ′′( x) < 0 if x > 2, f has inflection point ( 2,5 ) ,
lim f ( x) 8,=
lim f ( x) 0
The cost in dollars to manufacture x toys is given by
C ( x) =+
84 0.16 x − 0.0006 x 2 + 0.000003 x3
Find the marginal cost function C ′( x)
(c) Find C ′(100) and explain its meaning. What does it predict?
(c) Compare C ′(100) with the actual cost of producing the 101st toy.
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