The above statement is the false product rule! The correct product rule gives g (x) = 3x 4 cos x+ 12x 3 sin x. for all angles θ.

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1 Math 7A Practice Midterm III Solutions Ch. 6-8 (Ebersole,.7-.4 (Stewart DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual exam. You may expect to see problems on the test that are not exactly like problems you have seen before. True or False. Circle T if the statement is always true; otherwise circle F.. If g(x = x 4 sin x, then g (x = x cos x. T F The above statement is the false product rule! The correct product rule gives g (x = x 4 cos x+ x sin x.. sec θ tan θ = sin θ cos θ Notice that sec θ = cos θ for all angles θ. T F sin θ sin θ and tan θ =. So sec θ tan θ = cos θ cos θ cos θ = sin θ cos θ.. sin(t = sin t for all angles t. T F This is almost never true! For example, if you plug in t = π, on the left hand side you get ( π ( π sin =, whereas on the right hand side you get sin = =. So they are not the same! 4. tan ( π =. T F The reference angle for π (the angle in quadrant I corresponding to π is π. Using our special ( π triangle we get tan =. Since π is in quadrant II, and the tangent function is always ( π negative in quadrant II, we get tan =.. The only solution to the equation cos t = is t = π. T F Any angle that is coterminal with π also satisfies the equation. For example, t = π, t = ±π, t = ±π, etc. Multiple Choice. Circle the letter of the best answer.. The inverse of the function f(x = x 4 + is (a f (x = 4 x (b f (x = 4 x (c f (x = 4 x (d none of these; f(x does not have an inverse

2 f(x is a polynomial of degree 4; therefore it is not one-to-one and does not have an inverse.. 8 t 6 t = (a t (b t+ (c t (d 8t+ We have 8 t 6 t = ( t ( 4 t = t t = t+ t = t+ t = t+.. The inverse of the function f(x = x is f (x = x (a (b x (c x (d x To find the inverse of f(x we undo what has been done to x to get x by itself. Then we switch x and y (or you can switch x and y first and then solve for the new y, whichever you prefer. We have y = x y = x x = y 4. If ln(x = 4, then x = y = f (x = x. (a e4 + (b ln 4 + (c 4 + ln ln (d e 4/ + ln(x = 4 means e 4 = x. Solving for x we get x = e 4 + x = e4 +.

3 . If e x = 4 x, then x = (a ln ln 4 ln (b ln 8 (c + ln 4 + ln (d ln ln 4 The bases on each side are different, so we will need to pick a base and use logarithm laws. In light of the answer choices we should choose base e (i.e. natural log; we have ln ( e x = ln (4 x ln + ln ( e x = x ln(4 ln + x = (ln 4x x (ln 4x = ln ( ln 4x = ln x = ln ln 4. (be careful of the on the left side! 6. The inverse of the function f(x = 7 x is f (x = (a log 7(x + ( x (b log 7 (c log 7 (x + (d (x 7 We proceed as in #; we have y = 7 x log 7 (y = x (since log 7 ( undoes 7 log 7 (y + = x x = log 7(y + y = f (x = log 7(x + 7. The inverse of the function f(x = ln(x is f (x = (a e x (b ex (c ex (d ex/ +

4 Again we proceed as in #; we have 8. If f(x = tan x, then f (x = y = ln(x y = ln(x e y/ = x e y/ + = x x = ey/ + y = f (x = ex/ + (since e undoes ln( (a sec x (b sin x cos x (c tan x (d sec x tan x This is one of the formulas you should memorize. However, if you forget it you can remember that tan x = sin x and use the quotient rule to get the derivative, as we did in class. cos x 9. If f(x = x tan x, then f (x = (a sec x (b x sec x (c x sec x + tan x (d x sec Using the product rule, we have f (x = x sec x + tan x. 0. If f(x = tan( x + e x, then f (x = (a sec ( x + e x (b sec ( x + e x x + ex (c sec ( ( x + e x ( (d sec (x x + ex x + ex Using the chain rule with x + e x as the chunk we have f (x as in choice (c.. If f(x = 4 x, then f (x = (a (ln 4 4 x (b 4 x (c x 4 x (d ln(4 x

5 Another formula to memorize carefully. Or... here s a trick for deriving the formula: Notice that 4 x = e ln(4x = e x ln(4 (using logarithm laws and the fact that e undoes ln(. So the derivative is then e chunk (derivative of chunk = e x ln(4 (ln(4 = 4 x ln(4. This trick can come in handy if you forget the formula!. If f(t = (tan te t, then f (t = (a e t sec t (b e t tan t + e t sec t (c tan t sec (e t (d e t tan t + e t sec t Using the product rule (and the chain rule, we have f (t = tan t e t + e t sec t = e t tan t + e t sec t.. If f(x = ln(sin 7 x, then f (x = (a 7 cot x (b 7 ln(sin x (c sin 7 x (d 7 sin6 x cos x x First of all, recall that the derivative of ln(x is, so using the chain rule we have that the x derivative of ln(chunk is (derivative of chunk. chunk There are two ways to do this problem. One is to use a logarithm law to rewrite f(x in a simpler way: f(x = 7 ln(sin x. Then the derivative is f (x = 7 cos x = 7 cot x. sin x If you don t think of that, you can still get the answer using the chain rule; you will just have to do some more simplification at the end. We have f (x = sin 7 (x 7 sin6 (x cos x = 7 cos x = 7 cot x. sin x 4. If f(x = ln((x (x , then f (x = (a (b 70 x + 00x x (x (x (c x + 0 x (d (x + (x Similar to #, you can do this problem in two ways. However, it is much easier to use logarithm laws first, so that s what we ll do here. We have f(x = ln(x + 0 ln(x 6 + 7, so f (x = x + 0 x (0x = 70 x + 00x x If f(x = ln x x, then f (x =

6 (a x (b x x + x ln x (c (d x ln x ln x x 4 Using the quotient rule we have f (x = x x ln x x (x = x x ln x x 6 = x ( ln x x 6 = ln x x If $000 is invested at % interest compounded annually, the balance after 0 years is (a 000e.0 (c (b 000e 0 (d 000(.0 0 The formula for interest compounded annually is B(t = P ( + r t, where P is the principal or amount invested, r is the interest rate as a decimal, and t is the number of years. So we have B(0 = 000( = 000(.0 0.

7 Fill-In. Fill in the correct integer (positive, negative, or 0 whole number. (. log = =.. log(00 = 00 = 0.. e 4 ln 0 = 0000 e 4 ln 0 = e ln 04 = 0 4 = log 4 ( log 4 ( = Using a law of logarithms, log 4 ( log 4 ( = log 4 (/ = log 4 (4 =.. log 9 (007 = 007 We have log 9 (007 = 9 log 9 (007 = log /4 (64 = 64 = (. 4 Graphs. More accuracy = more points!. On the axes at right, sketch a graph of at least one period of the function f(t = sin(t π. The first thing to do is to rewrite the function as f(t = sin ( ( t π, so that we can see that the phase shift (horizontal shift is π to the right. The box (dashed lines shows where one period of the function goes. Notice that the amplitude is A =, the period is π = π, and there is no vertical shift.. For the graph of f(x shown at right, sketch a graph of f (x on the same axes.

8 The graph of the inverse of a function f(x can be obtained by reflecting the graph of f(x about the line y = x. You can also plot points by remembering that if (a, b is on the graph of f(x, then (b, a is on the graph of f (x. For example, the function f(x in this problem passes through the points (, and (, 0, so the inverse must pass through (, and (0,. Putting these ideas together, we get the graph shown. f(x y = x - f (x. On the axes below, sketch a graph of f(x = x+. Label at least two points on the curve. The graph of f(x can be obtained by shifting the graph of g(x = x to the left unit and down units. The graph at right shows both f(x and g(x. Since g(x = x passes through the points (, and (0,, we know the graph of f(x must pass through the points that are unit to the left and units down from these, namely (0, 0 and (,, respectively (-, x+ x - (0,0 4. On the axes below, sketch a graph of f(x = log (x +. Label at least two points on the curve. The graph is shown along with the graph of log x for reference. log x vertical asymptote (,- (,- (0,- log (x+ - (7,0 9 Work and Answer. You must show all relevant work to receive full credit.. Find the derivative of the function f(x = cos x e x4 +. Using the quotient rule (and the chain rule we have f(x = ex4 + (cos x / ( sin x cos x e x4 + 4x (e x4 + = ex 4 + sin x cos x 4x e x4 + cos x e (x4 +

9 . Find the inverse of the function f(x = x. We have y = x x = y x = y + x = y + Therefore f (x = x +. Simplify the expression log ( 7x 4 y+. ( 7x 4 ( We have log = log (7+log x 4 ( log y+ = +4 log x (y+ = 4 log x y + y+ 4. Solve for x: ln(x + = 6 ln x. We can get rid of the logarithms on both sides by applying e to both sides, but first we must get rid of the coefficients. Dividing both sides by we have ln(x + = ln(x ln(x + = ln x (this is a logarithm law e ln(x+ = e ln(x x + = x x x = 0 (x (x + = 0 x =, x = One of these, however, is not actually a solution to the original equation: x = cannot be plugged into ln x on the right side of the equation. So the only solution is x =. A population of a certain strain of bacteria doubles every 4 hours. How long does it take for the population to triple? If t is measured in hours, then the population of bacteria at time t is P (t = P 0 t/4, where P 0 is the initial population. If the initial population triples, then P (t = P 0. Therefore we have P 0 = P 0 t/4, or = t/4. Solving for t, we get log ( = t 4, so t = 4 log ( ( 8 hours 6. The half-life of 7 U (an isotope of uranium has a half-life of 6.7 days. How long does it take for a 00g sample to decay to 0g?

10 The amount of 7 U after t days is A(t = 00 have ( 0 = 00 ( 0. = log / (0. = ( t/6.7. We set A(t = 0 and solve for t; we t/6.7 t/6.7 t 6.7 t = 6.7 log / (0. (.4 days

True or False. Circle T if the statement is always true; otherwise circle F. for all angles θ. T F. 1 sin θ

True or False. Circle T if the statement is always true; otherwise circle F. for all angles θ. T F. 1 sin θ Math 90 Practice Midterm III Solutions Ch. 8-0 (Ebersole), 3.3-3.8 (Stewart) DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual exam.