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The final exam questions for a calculus course, covering limits, derivatives, integrals, and applications. Students are required to show their work for full credit. The exam consists of 10 problems, each worth 4 or 6 points.
Typology: Exams
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Name:
Final Exam
Show all your work to receive full credit for a problem. There are a total of 100 points on this test. Good luck!
(a) lim x→ 1 ex
(^3) −x
(b) lim x→− 3
x^2 − 9 x^2 + 2x − 3
(c) lim x→ 0
et^ − 1 t^3
(d) lim x→ 4 +
x − 4
(a) y = log 5 (1 + 2x)
(b) y = arctan(arcsin
x)
(c) y =
∫ (^) x
1
1 + t^4 dt
(d) x^2 y^2 + x sin y = 4
0.5 1.0 1.5 2.
(a) Partition the interval into four equal subintervals and use right approximating sums to compute an estimate of
0 f^ (x)dx. Explain, using the graph, what this sum represents.
(b) Write the sum in part (a) using sigma notation.
(c) Use the Fundamental Theorem of Calculus to calculate the exact value of the integral
0 f^ (x)dx.
(d) How could you change the sum in part (a) so that it’s closer to your answer in part (b)?
∫ (^) x
0
g(t)dt where g(t) is the function shown in the figure below. Answer the following questions about G.
1 2 3 4 5 6
1
2
(a) What is G(2)? G(3)?
(b) What is G′(1)? What is G′′(1)?
(c) Where are the stationary points of G?
(d) On what intervals is G increasing? On what intervals is G decreasing?
(e) What are the inflection points of G?
(f) On what intervals is G concave up? On what intervals is G concave down?
