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Exam 2 Review: 2.9-3.8This portion of the course covered the bulk of the formulas for differentiation, togetherwith a few definitions and techniques. Remember that we also left 2.9 for this exam.The following tables summarize the rules that we’ve had:f(x) f0(x) f(x) f0(x)c 0 cf cf0xnnxn−1f ± g f0± g0axaxln(a) f · g f0g + fg0exexfgf0g−fg0g2loga(x)1x ln(a)f(g(x)) f0(g(x))g0(x)ln(x)1xf(x)g(x)log diffsin(x) cos(x)cos(x) −sin(x)tan(x) sec2(x)sec(x) sec(x) tan(x)csc(x) −csc(x) cot(x)cot(x) −csc2(x)sin−1(x)1√1−x2tan−1(x)11+x2Vocabulary/Techniques:• Differentiable: A function f is differentiable at x = a if the limit exists. (Remember thedefinition of the derivative?) A function f is differentiable on (a, b) if it is differentiableat each point in the interval. Graphically, this means the function is smooth with novertical tangents.We showed that all differential functions are continuous, but not all continuous functionsare differentiable (consider y = |x|).• Implicit Differentiation: A technique where we are given an equation with x, y. Wetreat y as a function of x, and differentiate without explicitly solving for y first.Example: x2y +√xy = 6x → 2xy + x2y0+12(xy)−12(y + xy0) = 6• Logarithmic Differentiation: A technique where we apply the logarithm to y = f(x)before differentiating. Used for taking the derivative of complicated expressions, andneeded for taking the derivative of f(x)g(x).Example: y = xx→ ln(y) = x ln(x) →1yy0= ln(x) + 1 → ... etc• Differentiation of Inverses: If we know the derivative of f(x), then we can determinethe derivative of f−1(x). This technique was used to find derivatives of the inverse trigfunctions, for example:y = f−1(x) ⇒ f(y) = x ⇒ f0(y)y0= 1 From this, we could write:ddxf−1(x)=1f0(f−1(x))1Alternatively, we say that if (a, b) is on the graph of f and f0(a) = m, then we knowthat (b, a) is on the graph of f−1, anddf−1dx(b) =1m.NOTE: This is NOT the same as the derivative of (f(x))−1=1f(x), which isddx(f(x))−1= −(f (x))−2f0(x) =−f0(x)(f(x))2We also have an alternative version of the Chain Rule: For example, if Volume is afunction of Radius, Radius is a function of Pressure, Pressure is a func tion of time, thenwe can find the rate of change of Volume in terms of the Radius, or the Pressure, or thetime. Respectively, this is:dVdR,dVdP=dVdR·dRdP,dVdt=dVdR·dRdP·dPdt• Remember the logarithm rules:1. A = eln(A)for any A > 0.2. log(ab) = log(a) + log(b)3. log(a/b) = log(a) − log(b)4. log(ab) = b log(a)• Always simplify BEFORE differentiating. Example: To differentiate y = x√x, firstrewrite as y = x3/22Exam 2 Review Questions1. True or False, and explain:(a) The derivative of a polynomial is a polynomial.(b) If f is differentiable, thenddxqf(x) =f0(x)2√f(x)(c) The derivative of y = sec−1(x) is the derivative of y = cos(x).(d)ddx(10x) = x10x−1(e) If y = ln |x|, then y0=1x(f) The equation of the tangent line to y = x2at (1, 1) is:y − 1 = 2x(x − 1)(g) If y = e2, then y0= 2e(h) If y = |x2+ x|, then y0= |2x + 1|.(i) If y = ax + b, thendyda= x2. Find the equation of the tangent line to x3+ y3= 3xy at the point (32,32).3. If f(0) = 0, and f0(0) = 2, find the derivative of f(f(f(f(x)))) at x = 0.4. If f(x) = 2x + ex, find the equation of the tangent line to the inve rse of f at (1, 0).HINT: Do not actually try to compute f−1.5. Derive the formula for the derivative of y = cos−1(x) using implicit differentiation.6. Find the equation of the tangent line to√y + xy2= 5 at the point (4, 1).7. If s2t + t3= 1, finddtdsanddsdt.8. If y = x3− 2 and x = 3z2+ 5, then finddydz.9. A space traveler is moving from left to right along the curve y = x2. When she shuts offthe engines, she will go off along the tangent line at that point. At what point shouldshe shut off the engines in order to reach the point (4, 15)?10. A particle moves in the plane according to the law x = t2+ 2t, y = 2t3− 6t. Find theslope of the tangent line when t = 0. HINT: We can say thatdydx=dydtdxdt11. Find the coordinates of the point on the curve y = (x −2)2at which the tangent line isperpendicular to the line 2x − y + 2 = 0.12. For what value(s) of A, B, C does the polynomial y = Ax2+Bx+C satisfy the differentialequation:y00+ y0− 2y = x2Hint: If ax2+ bx + c = 0 for ALL x, then a = 0, b = 0, c = 0.313. If V = sin(w), w =√u, u = t2+ 3t, compute: The rate of change of V with respect tow, the rate of change of V with respect to u, and the rate of change of V with respectto t.14. Find all value(s) of k so that y = ektsatisfies the differential equation:y00− y0− 2y = 015. Find the points on the ellipse x2+ 2y2= 1 where the tangent line has slope 1.16. Differentiate. You may assume that y is a function of x, if not already defined explicitly.If you use implicit differentiation, solve fordydx.(a) y = log3(√x + 1)(b)√2xy + xy3= 5(c) y =qx2+ sin(x)(d) y = ecos(x)+ sin(5x)(e) y = cot(3x2+ 5)(f) y = xcos(x)(g) y =qsin(√x)(h)√x +3√y = 1(i) x tan(y) = y − 1(j) y =√x ex2(x2+ 1)10(Hint: Logarithmic Diff)(k) y = sin−1(tan−1(x))(l) y = ln |csc(3x) + cot(3x)|(m) y =−24√t3(n) y = x3−1/x(o) y = x tan−1(√x)(p) y = e2ex(q) Let a be a positive constant. y = xa+ ax(r) xy= yx(s) y =


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Whitman MATH 125 - Exam

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