, find the value(s) of a and b which make f differentiable at bx 2 + x if x 2 x = 2 or explain why no such values exist.

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1 Math 171 Exam II Summary Sheet and Sample Stuff (NOTE: The questions posed here are not necessarily a guarantee of the type of questions which will be on Exam II. This is a sampling of questions I have asked on previous exams over this material. These are simply provided as an additional source of practice.) DEFN indicates a definition, theorem, or concept to remember. FORM indicates a formula to remember. Section Power Rule (FORM) 2. Derivative of e x (FORM) 3. Exponential Solutions of Differential Equations 4. Derivatives of Piecewise Functions 5. Potential Proof Topic: Derivative Rules (+, -, scalar) a) Determine the points on the graph of f(x) = x(x + 1) whose tangent line also passes through the point (2, 3). b) Find the equation of the line tangent to the curve f(x) = 3x 2 at the point where x = 4. { x ax + 3 if x < 2 c) Given f(x) =, find the value(s) of a and b which make f differentiable at bx 2 + x if x 2 x = 2 or explain why no such values exist. Section Product Rule (FORM) 2. Quotient Rule (FORM) 3. Sample Problems: a) Find the derivative of f(x) = 2x x 1 b) Given f(2) = 4, f (2) = 3, g(2) = 1, and g (2) = 2, find u (2) if u(x) = (f(x) + 2x)(g(x)). c) Find the points where the graph of f(x) = x 2 e x has a horizontal tangent line. Section Limits Involving Trig Functions (FORM) 2. Derivatives of Trig Functions (FORM) 3. Trig Identities 4. Potential Proof Topics: a) Prove lim x 0 cos x 1 x = 0 b) Prove derivatives of the other trig functions given the derivatives of sin x and cos x. a) Find the equation of the line tangent to f(x) = 6x 2 cos x at the point where x = π 6. b) Differentiate y = sec(x) x 3 c) Compute lim x 0 sin(3x) cos x sin(7x)

2 Section Chain Rule (FORM) 2. Inside/Outside Functions 3. Multiple Chain Rule (work outside-in) 4. Potential Proof Topic: derivative of f(x) = a x a) Differentiate y = tan(sin x) b) Differentiate y = 2x2 + 3 sin 2 x c) Differentiate f(x) = x(x 2 + 1) 3 d) Suppose f and g are differentiable functions with the function values and derivatives given in the chart below: f(x) f (x) g(x) g (x) x = x = Section Implicit Differentiation 2. Tangent Lines to Implicitly-Defined Curves 3. Orthogonal Curves 4. Why you write y when differentiating implicitly (DEFN) 5. Potential Proof Topics: a) Proving Curves Orthogonal b) Formulas for inverse trig derivatives If h(x) = f(g(x)), what is h (2)? a) Given (y x) 2 = 2x + 4, find the equation of the tangent line at the point (6,2). b) Given sin(x + y) = xy + y, find y c) Compute f ( 1 2 ) if f(x) = tan 1 (2x 1) Section Derivative of ln x (FORM) 2. Logarithmic Differentiation 3. Derivatives of Logarithms in Other Bases (FORM) 4. Potential Proof Topic: Derivatives of ln x and log a x a) Differentiate f(x) = ln(2x 2 8) b) Find y if y = x sin x c) Compute f (e) if f(x) = ln(ln x)

3 Section 3.7SUPP 1. Definition of the derivative of a Vector Functions (DEFN) 2. Derivatives of Vector Functions (FORM) 3. Unit Tangent Vectors 4. Speed vs. Velocity 5. Vector/Parametric Equation of a Tangent Line (FORM from 1.3) 6. Potential Proof Topic: Prove the Rule for Differentiating a Vector Function 7. Sample Problems: Given the curve r(t) = (t 3 4t)i + (t 2 + 2t)j: a) If r(t) is the position of a particle at time t, find its speed at the point where t = 2 b) Find parametric equations of the line tangent to r(t) at the point where t = 1 (NOTE: this is NOT the same t value as part a!). c) Find a tangent vector of unit length to the curve r(t) =< t 2 + 1, t(2 t) > at the point (5, 8). d) Given r(t) = t i + t 2 j, find the acceleration when t = 1. Section 3.9SUPP 1. Derivatives of Parametrized Curves (FORM) 2. Tangent Lines to Parametrized Curves 3. Horizontal and Vertical Tangents 4. Given parameter vs. Given point (when vs. where) a) Find an equation of the line tangent to the curve r(t) = 2t + 3, t 2 + 2t at the point (5,3) b) Find the equation (in any form) of the line tangent to the graph of r(t) = t 3 3t 2, t 2 + 2t at the point corresponding to t = 1. c) Find the points on the curve x = t 2 + 4t, y = t 2 6t where the tangent line is horizontal or vertical (indicate which is which) Section Related Rates 2. Problem-Solving Strategies 3. Eliminating Extra Variables 4. Sample Problems: a) A ladder 26 feet long leans against a vertical wall (see figure below). The lower end of the ladder moves away from the wall at the rate of 5 ft/sec. How fast is the top of the ladder moving when the lower end is 10 feet from the wall?

4 b) A kite 80 ft above the ground moves horizontally at a speed of 8 ft/sec. When 100 feet of string have been let out, how fast is the string being let out, (i.e., how fast is the distance from the kite to the flyer changing)? c) An inverted conical tank has a small hole in the bottom and water is leaking out at a rate of 1 cubic inch per second. The tank is 10 inches tall and has a diameter across the top of 10 inches. How fast is the depth of the water changing when the depth is 6 inches? (the volume of a cone is V = 1 3 πr2 h) Section Linear Approximation of f at x = a (DEFN) 2. Differentials (FORM) 3. Significance of Linear Approxmation/Differentials (DEFN) 4. Potential Proof Topic: L(a) = f(a), L (a) = f (a) a) Given the function f(x) = x 3 4x 1, find the linear approximation of f at x = 2. Determine the x-intercept of the linear approximation. b) The radius of a 20cm tall cylindrical can is measured to be 10cm with a possible error of 0.5cm. Use differentials to estimate the maximum possible error in the volume of the can. The volume of a cylinder is V = π r 2 h. c) Use differentials or linear approximation to find an approximate value for Simplify any exponents. Section Mean Value Theorem (DEFN & FORM) 2. Increasing/Decreasing Functions (DEFN) 3. Concave Up/Concave Down Functions (DEFN) 4. Potential Proof Topics: a) If f > 0, then f inc (f < 0, then f dec) b) If f > 0, then f conc up (f < 0, then f conc down) c) Prove there is at most one solution to a given equation. c) If f = 0, then f is constant (or if f = g, then f(x) = g(x) + C)

5 Section 4.3, Implications of First Derivative (DEFN) 2. Implications of Second Derivative (DEFN) 3. Graph of Function Given Graph of Derivative 4. Graph of Derivative Given Graph of Function 5. Sketching Graph of Function given information about f, f, and f a) Given the graph below is the graph of the DERIVATIVE of a function f, determine the x-values for which f is increasing and decreasing, and the intervals where f is concave down and concave up b) Find the intevals where f is increasing and decreasing for the following functions: i) f(x) = x 3 2x 2 + x ii) f(x) = x 2 e 2x c) Find the inflection points for f(x) = x 2 ln x.