**MATH 140 Week 9
**

**Find the function from derivative and a single value.**

1. Find the antiderivative

1. Find the antiderivative

**a. **Given f’(x) = (5x^4)-(3x^2)+4 and f(-1)=2, find f(x)

**b.** Given f’(t)=t + (1/(t^3)) , t>0 , and f(1)=6 , find f(t)

**i. **Where f(t)=t+1/t^3 find as t + 1(numerator)/(t^3) (denominator)

**c.** Given f’(x) = (x+1)/(sqrt(x)) , and f(1) = 5 , find f(x).

i. Where f’(x) = (x+1 (as numerator))/(sqrt(x)denominator)).

**1**. Sqrt = Square Root Symbol

**2**. Symbols

a. Find f(x) if ((d^2)f)/(dx^2) = 6-18x , f(0)=6 , and f(2)=12

i. Where ((d^2)t) is numerator/(dx^2) is denominator

**Rectilinear Motion**

**3. A particle moves in a straight line and has acceleration given by a(t) = sin(t). Find the equation for the position of the particle s(t), given that x(pie/2) = 1 and s(0) = 0 (where v(t) is velocity at time t)**

**a.** Where v(pie(numerator)/2(denominator)

i. Pie is reference to Pi Symbol.

**4. A particle moves in a straight lune and has acceleration given by a(t) = sin(t) +cos(t). Find the equation for the position of the particle s(t) , given that v(0) = 4 and s(0) = 3 (where v(t) is velocity at time t ). **

Riemann Sums

**5. Integration**

**a.** Approximate the are under the curve y = x^2 +3 on [0,2] with a Riemann sum using 4 sub-intervals and midpoints.

**b. **How wude should each sub-interval be for approximating the are under the curve for y=2*(sqrt(2*x)) on [2,4] using 4 rectangles?

i. Where (2*x) is within the mouth of the Sqrt.

1. Sqrt = Square Root Symbol

**c.** Approximate the are under the curve for 2 * (sqrt(2*x)) on [2,4] with a Riemann sum using 4 sub-intervals and right enpoints.

i. Where (2*x) is within the mouth of the sqrt

1. Sqrt = Square Root Symbol

**d.** Approximate the area under the function f(x) , based ont eh given data, using left endpoints.

**6. A car starts moving at time t=0 and goes faster and faster. Its velocity is shown in the following table:**

Estimate how far the car travels during the 12 seconds.

**Definite integrals via geometry**

**7. Integration**

**a. **The sketh below represents the signed are of definite integral . Evaluare the integrak using an appropriate formula from geometry.

**b. **The sketch below represents the signed area of the definite integral . Evaluate the integral using and appropriate formula from geometry.

**c.** The sketch below represents the signed area of the definite integral . Evaluate the integral using an appropriate formula from geometry.

**8. Use geometry to compute the following definite integral:**

**9. Integration**

**a. **Given the following: