Calculus Answers

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1) The quantity of a substance can be modeled by the function k(t) that satisfies the differential equation dk/ dt =-1/90 (k-450). One point on this function is k(2) = 900. Based on this model, use a linear approximation to the graph of k(t) at t = 2 to estimate the quantity of the substance at t = 2.1.

2) Consider the differential equation dy/dx =x^2y^2with a particular solution y=f(x) having an initial condition y(−3) = 1 . Use the equation of the line tangent to the graph of f at the point (−3, 1) in order to approximate the value of f(−2.8) .

3) Given the differential equation dy/dx=-x/y^2 find the particular solution, y=f(x) , with the initial condition f(−6) = −3 .

4) What is the particular solution to the differential equation dy/dx =3 cos(x)y with the initial condition y(pi/2)=-2?

10. Find the values of k for which the function f(x) = −4x 2 + (4k − 1)x − k 2 + 4 is negative for all values of x.

Find the intervals of increase and decrease for the following functions: (a) f(x) = x 3 + 4x + 1

(b) f(x) = x 3 (5 − x) 2 .

(c) f(x) = x + sin x.

(d) f(x) = (x 2 − 4).

(e) f(x) = 2x 3 − 9x 2 + 12x

(f) f(x) = (x 2 − 4)2

(g) f(x) = x 3 − 2x 2 + x + 5 

Find the intervals of increase and decrease for the following functions: (a) f(x) = x 3 + 4x + 1 (b) f(x) = x 3 (5 − x) 2 . (c) f(x) = x + sin x. (d) f(x) = (x 2 − 4). (e) f(x) = 2x 3 − 9x 2 + 12x (f) f(x) = (x 2 − 4)2 (g) f(x) = x 3 − 2x 2 + x + 5 

8. Show that the real valued function defined by f(x) = x 3 − 3x 2 + 3x + 1 increases for all x ∈ R

6. Without evaluating derivatives, which of the following functions have the same derivatives? f(x) = ln x, g(x) = ln 2x, h(x) = ln x 2 , k(x) = ln 10x 2

5. Use the sign of the derivative to establish the inequaltiy ln(1 + x) > x − x 2 2 , ∀ x > 0

Use the Mean Value Theorem to establish the following inequalities (a) x < sin−1 x < x √ 1 − x 2 for 0 < x < 1. (b) e a (x − a) < ex − e a < ex (x − a) if a < x. (c) e x > x + 1, ∀ x > 0. (d) |sin x − sin y| ≤ |x − y| ∀x, y ∈ R. (e) x 1 + x < ln(1 + x) < x for −1 < x < 0 , x > 0 

3. Find an interval on which Rolle’s theorem applies to f(x) = x 3 − 7x 2 + 10x. Then, find all the critical points in that interval.

 (a) Does the function f(x) = x 3 satisfy the hypothesis of the Mean Value Theorem on the interval [−10, 10]? Justify your answer. (b) If so, determine the point c which is quaranteed by the Mean Value Theorem.