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This is the Exam of Calculus which includes Find, Differentiable, Function, Limit Definition, Derivative, Limits, Evaluate, Calculus, Respect, Elliptic Track etc. Key important points are: Differential Equation, Solves, Whether, Graphs, Related, Exists, Related, Limit De Nition, Derivative, Compute
Typology: Exams
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Your grade is based on correctness, completeness, and clarity on each exercise. Explain all answers completely. You may use a calculator, but no notes, books, or other students. Good luck!
1.) (10 pts.) Check whether y = ln x + x^2 + ex^ solves the differential equation y′′^ + xy′^ = 3.
2.) (15 pts.) Suppose that g(x) = f (x) + 3 and that f ′(x) exists for all x.
a.) Explain how the graphs of f and g are related.
b.) How is the graph of g′^ related to the graph of f ′? Explain.
c.) If f ′(1) = 5, what is g′(1)?
4.) (15 pts.) The graph below shows the function f. Use it to answer the questions about an antiderivative function F , and about f.
1 2 3 4 5 6 7 8 9 10 x
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f H x L
a.) At which x-value(s) does F have a local minimum?
b.) On which interval(s) is F increasing?
c.) On which interval(s) is F concave down?
d.) At which x-value(s) does F have an inflection point?
e.) On which interval(s) is f decreasing?
5.) (15 pts.) The graph below shows the function f. Use it to answer the questions below.
1 2 3 4 5 6 7 8 9 10 x
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f H x L
a.) Use the grid to estimate the values of f ′(1), f ′(5), and f ′(8).
b.) On which interval(s) is f ′′^ > 0?
c.) Using the axes below, sketch a graph of f ′.
1 2 3 4 5 6 7 8 9 10
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7.) (15 pts.) Use the graph of f (x) below to answer the questions about limits.
1 2 3 4 5 6 7 8 x
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f H x L
a.) What is lim x→ 2 f (x)?
b.) What is lim x→ 2 f ′(x)?
c.) What is (^) xlim→ 0 + f (x)?
d.) What is lim x→ 0 f (x)?
e.) What is (^) xlim→ 6 − f (x)?
BONUS: Write a poem about Calculus. You may use the back of this page or attach a page you have brought with you. (If you are attaching a page, please make sure your name is on it.)
