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5. Given: g(x) = ex tan x a) Which rule should be used to find the derivative? (Sum/difference, Product, or Quotient? b) Find g'(x). c) Evaluate g'(x) at x = 0. 6. Find f" (" iff(x) = secx Steps: . Find f' (x) . Find f"(x) in simplest form. . Substitute - into the expression for f"(x) and evaluate. x-+0 sin 6x 7. Find lim -by first writing the function in the form 1/reciprocal and using the special trig limit. 8. Find lim by first using a trig identity to replace tan x.MATH 2211 PRACTIS WORKSHEET (3.3) Calculus Topic: Derivatives of Trigonometric Functions Precalculus Topics: Evaluating and simplifying trigonometric function; using trigonometric identities 1. Multiply: a) sin x (sinx - cos x) b) sin x (2 cos x sinx ) 2. Use a trigonometric identity to simplify the expression: a) cosax sec x b) 3cos x 6 sin x 3. Multiply and simplify the result: a) (tan x + sec x) (tan x - sec x) b) (sin x + cos x) 2 4. Given: f(x) = 2 sinx + 3x a) Find f' (x). b) Evaluate f'(x) at x = "MATH 2211 PRAC T15 WORKSHEET [4.1) Calculus Topic: Maximum and Minimum Values Precalculus Topics: Solving Equations, Evaluating Functions Name: 1. Find the critical numbers for each of the following functions: Step 1: Find the derivative using the appropriate rule. If the derivative is undefined at any value of x, then that x is a critical number. Step 2: Set the derivative function equal to zero. Step 3: Solve for x. Any x that makes the derivative equal to zero is a critical number. Step 4: List all the critical x's. a} g(x) = ex(2 x) b] ') = sin I9 + cos 3, in the interval [0, 21!) c] y = (5x + 4)'2 MATH 2211 PRACTIS WORKSHEET (3.4) Calculus Topic: The Chain Rule Precalculus Topics: Composition of functions Name 1. For each composite function below, state the inner function u: a) f(x) = cos(x2) u = b) g(x) = (sin x) 3 u = c) H(x) = e-4x u: d) j(x) = tan(ex) u = e) K(x) = Vsecx u = x2+4 g) f(x) = 2(5x + 9)-4 2. State whether finding the derivative requires the chain rule. If yes, state the outer function y = f(u) and the inner function u = g(x). a) y = sin(3x) b) y = v2x - 7 c) y = sinx x d) y = (cotx) 2 e) y = e3x2-1 f) y = 2xe* g) y = esinxEach of the functions below is the composition of three functions, f. g, and h. We can say thaty = f(g(h(x))) wherey = f(u),u = g(w) and w = h(x). 3. f (x) = cos(e'2") y = u) = u=3(w)= f=( )( )( ) Simplify the derivative: 4. f (x) = 'jcos(4x + 1) y = u) = u = 9(w) = w = h(x) = d 3\" = ( ) ( )( ) Simplify the derivative: 5. {(1') = tan3 G) y=f(u)= u=g(w) = d =( )( )( ) Simplify the derivative: 2. TRUE or FALSE? A] If x = a is a critical number fory = f(x), then the graph must have a local maximum or local minimum at that point. B] If f'Ca) = 0, then we say that x = a is a critical number for y = f (x). C] Any function will attain an absolute maximum and an absolute minimum in a closed interval. 3. Given the function y = x3v'x 3 which is dened for all real numbers. a) Does the function have an absolute maximum point? b] Does the function have an absolute minimum point? c] lfwe consider the interval [1, 1], does the function have an absolute maximum point? :1] If we consider the interval [0, 5], does the function have an absolute minimum point? 4. Given: f(x) = 3x4 + 41:3 in the interval [2, 1} a) Find all critical numbers in the interval. b] Make a chart showing values of x and f(x) at the critical numbers and at the endpoints of the interval. c) The coordinates (x, y) of the absolute maximum are: ( , ) d] The coordinates (x, y) of the absolute minimum are: ( , )