Assignment:
Q1. Consider the function f(x)=(-5x^(2))+7x+15.
a) For the given function f, the expression f(x+h)-f(x)/h can be simplified to the form Ax+Bh+C, where A, B and C are constants. Do it.
Answer: f(x)=(-5x^2)+7x+15=____x+____h+____
b) Using the result of (a), find the derivative of f at x .
Answer: f'(x)=______. Note. The correct answer should be an expression in x.
c) Suppose that the position of a particle at time t is given by s=-5t^(2)+7t+15. What is the particle's instantaneous velocity at time t=2?
Answer: The particle's instantaneous velocity at time t=2 is _____
Q2. Differentiate the given functions. For this question, you may use the following differentiation rules and formulas:
If r is any rational number, then d/dx (x^r)=rx^(r-1).
If f and g are differentiable functions of x, and if α and β are constants, then
d/dx (αf(x)+ βg(x))= α d/dx (f(x))+ β d/dx (g(x)).
In each case, state the domain of the derivative.
a) f(x)=-3x^(3)-1x^(2)-4x+1.
Answer: The derivative of f(x) is f'(x)=______, and the domain of f' is ______
b)f(x)=5x^(3/2)-2x^(1/2)-1x^(-1/2)-4x^(-3/2).
Answer: The derivative of f(x) is f'(x)=_____, and the domain of f' is _____
c) f(x)=-3sqrt(x)+2/sqrt(x)+2/x+_3/x^(2).
Answer: The derivative of f(x) is f'(x)=______, and the domain of f' is ______
Q3. Differentiate the following functions and simplify your answers into the specified form.
a) f(x)=-1x-7/-10x^(2)-7.
Answer: f'(x)=1/(-10x^(2)-7)^2 (____x^(2)+_____x+____)
b) g(x)=x^(3)/-9x+6
Answer g'(x)= 1/(-9x+6)^2 (____x^(3)+_____x^(2)+_____x+_____)
Q4. Differentiate the following functions and simplify your answers into the specified form.
a) f(x)=(12x-8)sinx+(-7x-5)cosx.
Answer: f'(x)=(_____x+_____)sinx+(_____x+_____)cosx. Note: The four answers should be constants
b) g(x)=(5x^(2)-8)tanx.
Answer: g'(x)=(_________)sec^(2)x+(___________)tanx. Note: The two answers shoud be functions of x
Provide complete and step by step solution for the question and show calculations and use formulas.