Math 113 Final Exam Practice

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1 Math Final Exam Practice The Final Exam is comprehensive. You should refer to prior reviews when studying material in chapters 6, 7, 8, and.-9. This review will cover.0- and chapter 0. This sheet has three sections. The first section will remind you about techniques and formulas that you should know. The second gives a number of practice questions for you to work on. The third section gives the answers of the questions in section. Review Finding sums of series - Additional information Finding a power series that represents a specific function is the next topic. The first one we learned was the geometric series: x = x n iff x (, ). We then found the sum of several series by differentiating, integrating, multiplying by x, etc. The Taylor series of a function is f (n) (c) (x c) n and can also be used to find the power series of a function. Notice that the interval of convergence of these series is still very important. We need to know when we can trust them. In addition to the geometric series above, the following Maclaurin series (with interval of convergence) are important: tan x = ln( + x) = e x = ( ) n x n+, [, ] n + ( ) n x n+, (, ] n + x n sin x = cos x = sinh x = ( ) n x n+ (n + )! ( ) n x n (n)! x n+ (n + )! ( ) r ( + x) r = x n x n, (, ) where the binomial cosh x = n coefficients are ( (n)! ) r n = r(r ) (r n+) If you need to construct a Maclaurin series of a function and some of the above functions are included, it is almost always easier to manipulate the Maclaurin series instead of constructing the series by scratch. Approximating sums of series In addition to finding whether sums of series converge or not, we also were able to find approximations to the error. There were basic approximations to the error given by the Integral test, Alternating Series test, and the Taylor Series.. If a k is convergent with sum s and f(k) = a k where f is a continuous, positive, and decreasing function for x n, then the remainder R n = s s n = n+ k=n+ a k satisfies the inequality f(x) dx R n n f(x) dx. If {a k } is a positive decreasing sequence with a limit of 0, then ( ) k a k is convergent with sum s and the remainder R n = s s n = k=n+ ( ) k a k satisfies the inequality R n < a n+

2 . Taylor s Inequality: If T n (x) = n k=0 the remainder R n (x) = f(x) T n (x) satisfies the inequality on the interval where f (n+) (x) < M. f (k) (c) k! (x c) k is the nth Taylor polynomial of f(x) centered at c, then R n (x) M x c n+ (n + )! We use this information, when applicable, to find maximum errors when approximating a function by a Taylor polynomial as well. 0. Parametric Equations We learned how to define curves parametrically. That is, we learned how to describe a curve given by an equation in terms of a pair of functions You will need to be able to do the following: (a) Graph a curve from its parametric equations. (b) Recognize the curve of a set of parametric equations. H(x, y) = 0 x = f(t), y = g(t). (c) Eliminate the parameter of the parametric equations to find an equation in x and y describing the curve. (d) Construct a set of parametric equations for a curve written in cartesian coordinates. 0. Calculus of Parametric Equations In the discussion below, we will assume that a curve can be described parametrically by Slopes dy dy dx = dt x = f(t), y = g(t). This gives a formula for the slope as a function of parameter. Arclength Surface Area Rotated about the x axis: Rotated about the y axis: Area under the curve Area between the curve and the x axis: Area between the curve and the y axis: S = S = s = t t A = A = t dx dt d y dx = d dt = g (t) f (t). ( dy dx dx dt ) (f (x)) + (g (x)) dx πf(x) (f (x)) + (g (x)) dx πg(x) (f (x)) + (g (x)) dx t y dx = t x dy = t t g(t)f (t) dt f(t)g (t) dt

3 0. Polar Coordinates In this section, we learned how to write points and equations in polar coordinates. You will need to be able to do the following: (a) Convert a point from cartesian coordinates to polar coordinates and vice-versa. (b) Be able to write a polar curve in cartesian coordinates and a cartesian curve in polar coordinates. (c) Be able to graph and recognize polar curves. You will need to know the following formulas: x = r cos θ y = r sin θ r = x + y θ = tan (y/x) if (x, y) is in quadrants or 4. Otherwise, θ = tan (y/x) + π. You will also need to be able to find the slope of a polar curve. Fortunately, we can do this using the standard parameterization of a polar curve. If r = f(θ) is a polar curve, then from the above equations we can write x = f(θ) cos θ y = f(θ) sin θ. Then we can use the techniques of section 0. to find the slope of the tangent line. 0.4 Calculus of Polar coordinates Note that we can use the parameterization of polar curves mentioned in the previous section to find arclength also. However, in this case, the formula simplifies considerably, so it is better to use the simplified formula directly. If r = f(θ), then the arclength is given by s = θ θ 0 (f(θ)) + (f (θ)) dθ. We also wish to find area underneath polar curves. However, since polar curves are defined by angle, underneath really translates to between the curve and the origin. The area inside a polar curve r = f(θ), or between the polar curve and the origin is given by θ A = θ 0 f (θ) dθ.

4 Questions Try to study the review notes and memorize any relevant equations before trying to work these equations. If you cannot solve a problem without the book or notes, you will not be able to solve that problem on the exam.. Find the Maclaurin series for f(x) = ln( x) from the definition of a Maclaurin series. Find the radius of convergence.. Find a Taylor series for f(x) = cos(πx) centered at x =. Prove that the series you find represents cos(πx) for all x.. Use multiplication to find the first 4 terms of the Maclaurin series for f(x) = e x cosh(x). 4. Use division to find the first terms of the Maclaurin series for g(x) = x cos x. 5. Use the power series of +x to estimate. correct to the neares.000. Justify that the error is less than using the Alternating Series Estimation Theory or Taylor s Inequality. 6. Find the sum: (a) (b) (c) ( ) n ( ) n+ (n+) n n= ( ) n x n (n+)! (d) 4! + 8! + 6 4! + 5! Find the Taylor polynomial T (x) for the function f(x) = arcsin x, at a = Approximate f by a Taylor polynomial with degree n at the number a. And use Taylor s Inequality to estimate the accuracy of the approximation f(x) T n (x) when x lies in the given interval. (a) f(x) = x, a = 8, n =, 7 x 9 (b) f(x) = x sin x, a = 0, n = 4, x 9. Find the Taylor polynomial T (x) for the function f(x) = cos x at the number a = π/. And use it to estimate cos 80 0 correct to five decimal places. 0. A car is moving with speed 0m/s and acceleration m/s at a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Would it be reasonable to use this polynomial to estimate distance traveled during the next minute?. Show that T n and f have the same derivatives at a up to order n. In problems to 4, graph the parametric curve.. x(t) = cos t, y(t) = sin(t).. x(t) = e t, y(t) = ln(t) x(t) = t, y(t) = t / t. In problems 5 to 7, eliminate the parameter to find a Cartesian equation of the curve. 5. x(t) = cos t, y(t) = sin(t). 6. x(t) = e t, y(t) = ln(t) x(t) = t, y(t) = t / t. In problems 8 to 9, find parametric equations for the curve 8. x + y 4 = 9. y = x + x 0. Find an equation of the tangent to the curve at the given point. x = cos(θ)+sin(θ), y = sin(θ)+cos(θ); θ = 0. For which values of t is the tangent to curve horizontal or vertical? Determine the concavity of the curve. x = t t, y = t 6t. Find the area enclosed by the curve x = t t, y = t and the y-axis.. Find the area of one quarter of the ellipse described by x = 5 sin(t), y = cos(t). 4. Find the exact length of the curve: x = t +t, y = ln( + t); 0 t. 5. Find the exact length of the curve: x = e t +e t, y = 5 t; 0 t. 6. Find the exact surface area by rotating the curve about the x-axis: x = t, y = t ; 0 t. 7. The Cartesian coordinates for a point are (, ). Find polar coordinates (r, θ) for the point where r > 0 and 0 θ < π. 8. Find the distance between the points with polar coordinates (, π/) and (4, π/). 9. Find a polar equation for the curve represented by the Cartesian equation x + y = Identify the curve given in polar coordinates by r = 4 sin θ by finding a Cartesian equation for the curve.. Graph in polar coordinates r = cos(θ).. Graph in polar coordinates r = sin(θ).. Graph in polar coordinates r = + cos θ.

5 4. Graph in polar coordinates r = cos(θ). 5. Find the area inside the circle r = 6 sin θ and outside the limaçon r = + sin θ. 6. Find the area of one petal of the rose given by r = cos θ. 7. Find the length of the polar curve given by r = θ for θ [0, π]. 8. Set up but do not evaluate an integral for the length of the polar curve given by r = θ + sin θ for θ [0, π ]. Answers. ln( x) =. cos(πx) = R n (x) πn+ x n+ (n+)! f (n) (0) x n = ln + f (n) ()(x ) n = 0 for all x x n n n n= : (R = ) ( ) n+ π n (x ) n (n)!.. e x cosh x = ( + x + x (x)! +...)( +! + ) = + x + 5 x + 6 x + 4. x cos x = x x! + x4 4! x = 6 x x = x + x 9 4x 8 + Thus, Since the series is alternating the error for this sum is less than the size 7 of the next term, which is 40500, which is less than Find the sum: ( ) (a) n ( = ) n+ (n+) tan ( ) = π 6 (b) (c) n = e ( ) n x n (n+)! = sin x x (d) 4! + 8! + 6 4! + 5! +... = e 7. T (x) = x + 6 x 8. (a) + x 8 (x 8) 88 : R f () (7)! (b) x x4 6 : R 4 f (5) () 5 5! (x π ) + 6 (x π ) : cos 80 = cos 4π T (x) = s(0) + s (0)x + s (0) x = 0x + x T () = m: No. Prove by mathematical induction or directly consider the k th derivative of the polynomial T n y = x x 6. y = ln(ln(x)) ln() + 7. y = x x 8. x = cos(t), y = sin(t) 9. x = t, y = t + t 0. y = x. vertical at t = /, horizontal at ±, concave up when t > /, concave down when t < / π 4. 0/ + ln( + 0) + ln( + ). 5. e e 6. 5 π( ).

6 (, 4π/) π/ Convert (, π/) to Cartesian coordinates. x = cos(π/) =, y = sin(π/) = Convert (4, π/) to Cartesian coordinates. x = 4 cos(π/) =, y = 4 sin(π/) = Find the distance between (, ) and (, ) in Cartesian coordinates. 9 + = = 9. r =. 0. r = 4 sin θ gives r = 4r sin θ. In Cartesian coordinates x + y = 4y or x + y 4y + 4 = 4 or x + (y ) = 4 This is a circle of radius centered at (0,) r = 6 sin θ and r = + sin θ intersect at θ = π 6 and θ = 5π 6. Use symmetry to get the integral: A = π/ π/6 ( (6 sin θ) ( + sin θ) ) dθ = 4π 6. One petal is drawn for θ [ π 6, π 6 ]. Use symmetry to get the integral: A = π/6 0 cos θ dθ = π 7. The length of the curve is the integral:. L = π 0 θ + dθ = π π + + ln (π + π + ) 8. The length of the curve is the integral:. L = π/ 0 (θ + sin θ) + ( + cos θ) dθ

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You can learn more about the services offered by the teaching center by visiting MAC 232 Exam 3 Review Spring 209 This review, produced by the Broward Teaching Center, contains a collection of questions which are representative of the type you may encounter on the exam. Other resources