MAT131 Fall 2008 Midterm 2 Review Sheet

The topics tested on Midterm 2 will be among the following.

(i) Given a function, its first derivative and its second derivative, use the first derivative test to determine where the function is increasing and where decreasing, and use the second derivative test to determine where the function is concave up and where concave down. Combine this with even/odd, identification of vertical and horizontal asymptotes, and identification of discontinuities to give a rough sketch of the graph of the function.

(ii) Use the rules of differentiation: the sum rule, the product rule, the power rule and the derivatives of exponentials.

(iii) Given the limits lim_{h→0} \frac{sin(x)}{x} = 1 and lim_{h→0} \frac{1 − cos(x)}{x} = 0, use trigonometric identities to find the formulas for the derivatives of sin(x) and cos(x) as limits of difference quotients.

(iv) Find the derivatives of other trigonometric functions – tan(x), cot(x), sec(x) and csc(x) – using the derivatives for sin(x) and cos(x) and the rules for differentiation.

(v) Find derivatives using the chain rule.

(vi) Find the tangent slope to a parametric curve at a specified point.

(vii) Find derivatives using implicit differentiation, including derivatives of inverse functions.

(viii) Find derivatives using logarithmic differentiation.

(ix) Find the linear approximation to the value of a function, using a known nearby value and derivative or using differentials. Determine whether your approximation is an overestimate or an underestimate.

(x) Understanding differential notation and the geometric interpretation of differentials. Using differentials to approximate values of functions.

Following are some practice problems. More practice problems are in the textbook as well as on the practice midterm.

Problem 1. In each of the following cases, a function y = f(x) is given as well as the first derivative f′(x) and the second derivative f′′(x). In each case do all of the following.

(i) Compute the first and second derivatives of f(x) to verify the given formulas.

(ii) Determine whether f(x) is even, odd or neither.

(iii) Find the equations of all vertical and horizontal asymptotes.

(iv) Determine where y = f(x) is increasing and where y = f(x) is decreasing. Pay careful attention to the change as x crosses a vertical asymptote.

(v) Determine where y = f(x) is concave up and where y = f(x) is concave down. Pay careful attention to the change as x crosses a vertical asymptote.

(vi) Give a rough sketch of the graph of y = f(x) carefully labelling all of the above information on your sketch.

(a)

f(x) = x^{3} − x,
f′(x) = 3x^{2} − 1,
f′′(x) = 6x.

(b)

f(x) = x^{4} − 2x^{2},

f′(x) = 4x^{3} − 4x,
f′′(x) = 12x^{2} − 4.

(c)

f(x) = x^{−1}e^{x} = \frac{e^{x}}{x} ,

f′(x) = (x − 1)x^{−2}e^{x} = \frac{(x − 1)e^{x}}{x^{2}} ,

f′′(x) = (x^{2} − 2x + 2)x^{−3}e^{x} = \frac{(x^{2} − 2x + 2)e^{x}}{x^{3}} .

(d)

f(x) = x^{−2}e^{x} = \frac{e^{x}}{x^{2}} ,

f′(x) = (x − 2)x^{−3}e^{x} = \frac{(x − 2)e^{x}}{x^{3}} ,

f′′(x) = (x^{2} − 4x + 6)x^{−4}e^{x} = \frac{(x^{2} − 4x + 6)e^{x}}{x^{4}} .

(e)

f(x) = x^{−3}e^{x} = \frac{e^{x}}{x^{3}} ,

f′(x) = (x − 3)x^{−4}e^{x} = \frac{(x − 3)e^{x}}{x^{4}} ,

f′′(x) = (x^{2} − 6x + 12)x^{−5}e^{x} = \frac{(x^{2} − 6x + 12)e^{x}}{x^{5}} .

(f)

f(x) = x^{−1} ln(x^{2}) = \frac{ln(x^{2})}{x}

f′(x) = x^{−2}(2 − ln(x^{2})) = \frac{2 − ln(x^{2})}{x^{2}},

f′′(x) = x^{−3}(2 ln(x^{2}) − 6) = \frac{2 ln(x^{2}) − 6}{x^{3}}.

(g)

f(x) = x^{−2} ln(x^{2}) = \frac{ln(x^{2})}{x^{2}},

f′(x) = x^{−3}(2 − 2 ln(x^{2})) = \frac{2 − 2 ln(x^{2})}{x^{3}},

f′′(x) = x^{−4}(6 ln(x^{2}) − 10) = \frac{6 ln(x^{2}) − 10}{x^{4}}.

(h)

f(x) = (1 + x^{−2})^{1/2} = \frac{1}{\sqrt{1 + (1/x)^{2}}} ,

f′(x) = −x^{−3}(1 + x^{−2})^{−1/2} = \frac{−1}{x^{3}\sqrt{1 + (1/x)^{2}}} ,

f′′(x) = (3x^{2} + 2)x^{−6}(1 + x^{−2})^{−3/2} = \frac{3x^{2} + 1}{x^{6}(\sqrt{1 + (1/x)^{2}})^{3}} .

Problem 2. What is the differential of y = x^{3}? When x = 2, a small change dx = 0.01 in x produces what change dy in y?

Problem 3. The inverse function of

f(x) = \frac{1}{2}(e^{x} + e^{−x})

is called the inverse hyperbolic cosine, cosh^{−1}(x). For y = cosh^{−1}(x), find a formula for the derivative of y that is an expression only involving polynomials in x and square roots.

Hint. After you find an answer that involves y, consider what you get by squaring your formula for y′ and squaring your formula for x. Can you relate these two formulas?

Problem 4. The double-angle formula for tangent is

tan(2x) = \frac{2 tan(x)}{1 − (tan(x))^{2}} .

Compute the derivative of each side of this equation. Which derivative is easier to compute?

Problem 5. Let a be a positive constant and consider the parametric curve

\{ \begin{matrix}x(t) & = & 2at \\ y(t) & = & \frac{2a}{1+t^{2}} \end{matrix}

Compute the slope of the tangent line at the point where t = 1/\sqrt{3}. Problem 6. Compute each of the following derivatives. (a)

y = ln(ln(x)), x > 1

(b)

y = e^{e^{x}}

(c)

y = \frac{2x}{1 + x^{2}}

(d)

y = \frac{x^{3}\sqrt{sin(x)}}{\sqrt{cos(x)}}

(e)

y = ln(−x + \sqrt{x^{2} − 1})

(Simplify your answer as much as possible.) Problem 7. Using your known formulas for the derivatives of sin(x) and cos(x), find the limit

lim_{h→0} \frac{cos(h) − 1}{h}

by relating it to the derivative of cos(x) for some value of x. Problem 8. The value of 1/(1 + \sqrt{10}) is close to 1/4 = 0.25. Using a linear approximation or differentials, estimate whether the true value is closer to 0.2 or closer to 0.3. Problem 9. For the curve with implicit equation

y + \frac{1}{y} = 4x + 2x^{2}

find the slope of the tangent line to the curve at the point (x, y) = (1/2, 2).

Problem 10. Find the derivative y′ of the function y in each of the following cases.

(a)

y = sin(\sqrt{x^{2} + 1})

(b)

y = (ln(x^{2}))^{2}

(c)

y = (\sqrt{x})^{cos(x)}

(d)

y = \frac{x^{2}e^{x}}{x^{2} + 2x + 1}

(e)

x^{2} + 4y^{2} = 5

(f)

y = \frac{(x^{2} + 1)^{3}}{x^{2} − 1}

Problem 11 In each of the following two cases, find the linearization of f(x) near the point x = a.

(a)

f(x) = x^{2/5}, a = 32.

(b)

f(x) = \frac{1}{\sqrt{1 + x^{2}}} , a = \sqrt{3}.

Problem 12 Using differentials or an appropriate linear approximation, approximate the following number. Also say whether the approximation is an overestimate or an underestimate.

\frac{1}{\sqrt{25.1}}