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The directions and problems for a calculus test, including finding derivatives, integrals, and solving differential equations. Topics covered include trigonometric functions, logarithmic functions, exponential functions, and inverse functions.
Typology: Exams
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You must show your work on this test paper. Do not use scrap paper. When you use your calculator, indicate how you use it, e.g., “I used my calculator to find y′^ = 2x.”
d dx (sin−^1 (x^2 y)). An answer alone is sufficient.
dQ dt
= 300 − 0. 3 Q. Solve the differential equation subject to Q(0) = 500.
3 x − 7 5 x + 11
. Use calculus to argue that f is invertible.
12 x + 13. Find g′(23) if g = f−^1.
(a) y′′^ = y (I) y = cos x (b) y′^ = −y (II) y = sin( 2x) (c) y′^ = 1/y (III) y = x^2 (d) y′′^ = −y (IV) y = ex^ + e−x (e) y′′^ − 4 y = 0 ( V)y =
2 x
Time Weight (in grams) 9 am 10. 10 am 8. 11 am 8. 12 noon 7. 1 pm 6.
(a) (7 pts) Determine a formula of the form Q = Q 0 e−kt^ which would give the weight of the substance, Q, at time t, where t is measured in hours since 9 am. Use your calculator, but indicate how you get to your answers. Give your numbers to three decimal places.
(b) (8 pts) Find the half-life of this substance, accurate to three decimal places.
(a) (10 pts) Write a differential equation to model the population as a function of time. (Hint: Recal that r.g.r. = k − aP and that the logistic model is
dP dt
= k − aP .)