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- Find derivatives of the functions defined as follows. y=-4e-0.3xCalculate the difference quotient for f (x) = 5 – x² at a = 3. (Express numbers in exact form. Use symbolic notation and fractions where needed.) f (x) – f (3) х — 3 f (x) – f (3) Calculate f' (3) by finding lim x→3 3 (Give an exact answer. Use symbolic notation and fractions where needed.) f' (3) = Find an equation of the tangent line to f (x) at a = 3. (Express numbers in exact form. Use symbolic notation and fractions where needed. Let f (x) = y and express equation in terms of x and y.)2. a) Evaluate the following: -1 x-1(x+1)' xả-5x-20 x-5 x²-4x-5' x-5 i. lim ii. lim iii. lim- x-5x²-10x+25 √x²+4-2 x² 5h+4-2 iv. lim x-0 V. lim- h→0 h b) Find the derivative of the following functions: i. f(x) = -secx + 1 - secx ii. f(x) = x-5 tan 4x + sin 4x x iii. f(x) = ln(x-¹) [sinx + √x² + 1]. iv. f(t) = 3t² cost - t³ sin √t, e2x In(x²-3x+8) v. f(x) = (x+1)²
- Sketch the graph of a function y = f(x) with all of the following properties. a. f'(x) > 0 for -9 sx 0 for x > 2 d. f(2) = -2 and f(0) lim f(x) = 0 and lim f(x) : = 1 е. = 00 X-00 X- 00 f. f'(1) does not exist.Suppose f(x)=8-x². (a) Find the slope of the tangent line to the graph y = f(x) when x = -1. f(x+h)-f(x) h -2-2h Slope = lim h→0 = lim h→0 = = 2 (b) Find an equation for the tangent of to the curve y = f(x) at the point (-1,7). Tangent line: y = 2x+9 (c) On a piece of paper, sketch the curve y = f(x) and the tangent line together.90. The function C(t) = K(ea – eb), where a, b, and K are positive constants and b> a, is used to model the concen- tration at time t of a drug injected into the bloodstream. (a) Show that lim, C(t) = 0. (b) Find C"(t), the rate of change of drug concentration in the blood. (c) When is this rate equal to 0?
- Suppose f(x) = 13 – x² (a) Find the slope of the tangent line to the graph y = f (x)x = −1 when lim h→0 Slope = = f(x+h)-f(x) h Tangent line: - (² = : lim h→0 (b) Find an equation for the tangent of to the curve y = f(x)(−1, 12) at the point y 2x+14 -2x = 2b. Determine if the lim f(x) exists, if f(x) is defined as follows: (2x + 3, 15x-9, x < 4 x 24 f(x) = c) Find the first derivative of the following functions: i) y = 3x' + 4x -5 ii) y = x'exIf f(x) is a differentiable function such that f'(0) 2, f'(2) = = -3 and f'(5) = 7, then the lim ()-7e) is equal to (f(x)-f(2) а. 2 b. 3 С.- 3 d. - 2
- The equation of a tangent line at a point (x₁, f (x₁)) where limh→0 f(x₁+h)-f(x₁) h = +∞ is x = x1. O True O FalseCalculate the difference quotient for f(x) = 5 – x² at a = 4. (Express numbers in exact form. Use symbolic notation and fractions where needed.) f(x) – f(4) x – 4 f(x) – f(4) Calculate f' (4) by finding lim x→4 x – 4 (Give an exact answer. Use symbolic notation and fractions where needed.) f'(4) = Find an equation of the tangent line to f(x) at a = : 4. (Express numbers in exact form. Use symbolic notation and fractions where needed. Let f(x) = y and express equation in terms of x and y.) equation:1. Find f'(3) using f'(a) = lim h→0 f(x) = 2x² + 3x - 5 f(a+h)-f(x) h for the following function