# Techniques - Calculus - Exam, Exams for Calculus. Jaypee University of Engineering & Technology

## Calculus

Description: These are the Exam of Calculus which includes Worst, Wedding Cake, Volume of Cylinder, Very Cold Freezer etc. Key important points are: Techniques, Derivatives, Valid Techniques, Enough Steps, Clear, Integral, Interpret, Geometry, Compute, Signed Area
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MATH 105 FINAL EXAM April 13, 2010
Name:
exercise. You may use a calculator, but no notes, books, or other students.
Good luck!
1.) (15 pts.) Compute the following derivatives, using any valid techniques you like. (Be
sure to show enough steps so that it is clear which techniques you are using.)
a.) (5 pts.) f(x) = π+extan x
b.) (5 pts.) p(t) = 1
t+1
1 + t2
c.) (5 pts.) y=xcos x
1
2.) (15 pts.) Consider the integral Z2
1
(x+ 1) dx.
a.) (5 pts.) Interpret the integral as “area under a curve”. Draw the graph with area
shaded in, and use geometry to compute the signed area.
b.) (5 pts.) Evaluate the integral using the Fundamental Theorem of Calculus.
should not, be the same.
2
3.) (15 pts.) Consider the integral Z2
1
1
x2dx.
a.) (5 pts.) Approximate the area under the curve using the a trapezoid sum with two
equal subintervals. In your approximation, use at least two digits after the decimal
point.
b.) (5 pts.) Evaluate the integral using the Fundamental Theorem of Calculus.
should not, be the same.
3
4.) (20 pts.) Use the graph of f(x), below, to complete the following questions.
-5
-4
-3
-2
-1
1
2
3
4
5
-5
-4
-3
-2
-1
1
2
3
4
5
fHxL
-5
-4
-3
-2
-1
1
2
3
4
5
-5
-4
-3
-2
-1
1
2
3
4
5
f¢HxL
a.) At which x-value(s)
i.) (2 pts.) does F, an antiderivative of f, change from concave up to concave down?
ii.) (2 pts.) does F, an antiderivative of f, have a stationary point?
iii.) (2 pts.) does F, an antiderivative of f, have a local maximum?
iv.) (2 pts.) is fgreatest?
v.) (2 pts.) is f0greatest?
vi.) (2 pts.) is F, an antiderivative of f, greatest?
b.) (2 pts.) What kind of symmetry does fhave?
c.) (4 pts.) Sketch the derivative f0(x) on the given set of axes.
d.) (2 pts.) What kind of symmetry does f0have?
4