Calculus mixed test 2

1. 1
   Bound sine

   The absolute value of sine never exceeds one.

   $$|\sin(x^{-2})|\le1$$
2. 2
   Bound the product

   Multiply by the absolute value of x.

   $$|x\sin(x^{-2})|\le|x|$$
3. 3
   Squeeze

   The bound approaches zero.

   $$\lim_{x\to0}x\sin(x^{-2})=0$$

Independent check: Every sampled absolute value is no larger than |x|, so the envelope converges to zero.

1. 1
   Normalize

   Use the highest denominator power.

   $$\frac{5/x-2/x^4}{1+7/x^4}$$
2. 2
   Take the limit

   All reciprocal powers approach zero.

   $$\frac{0-0}{1+0}=0$$

Independent check: For large positive x, the expression behaves like 5/x, which approaches zero.

1. 1
   Differentiate implicitly

   Apply the chain rule to y squared.

   $$2x+2yy'=0$$
2. 2
   Solve for the slope

   Divide by 2y.

   $$y'=-\frac{x}{y}$$
3. 3
   Evaluate

   Insert the coordinates.

   $$y'=-\frac34$$

Independent check: The radius from the origin has slope 4/3, and the tangent is perpendicular with slope -3/4.

1. 1
   Differentiate in time

   Radius depends on time.

   $$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$$
2. 2
   Substitute

   Use the stated radius and growth rate.

   $$4\pi(3)^2(2)=72\pi$$

Independent check: Using r(t + h) = 3 + 2h, the quotient [V(3 + 2h) - V(3)]/h approaches 72π, independently confirming the rate.

1. 1
   Build the linearization

   Evaluate the function and derivative at 4.

   $$f(4)=2,\quad f'(4)=\frac14$$
2. 2
   Approximate

   The input change is 0.1.

   $$L(4.1)=2+\frac14(0.1)=2.025$$

Independent check: Squaring 2.025 gives 4.100625, close to 4.1; concavity predicts a slight overestimate.

1. 1
   Integrate by parts

   Differentiate x and integrate the scaled exponential.

   $$\int xe^{2x}dx=\frac{x}{2}e^{2x}-\frac14e^{2x}+C$$
2. 2
   Evaluate

   Compute upper minus lower.

   $$\left[\frac{e^{2x}(2x-1)}4\right]_0^1=\frac{e^2+1}{4}$$

Independent check: Differentiating the antiderivative returns xe^(2x), and numerical quadrature agrees with the endpoint value.

1. 1
   Use an antiderivative

   Integrate each term.

   $$F(x)=\frac{x^3}{3}-x$$
2. 2
   Evaluate

   Subtract the lower endpoint value.

   $$F(1)-F(-1)=-\frac23-\frac23=-\frac43$$

Independent check: One-panel Simpson's rule is exact for this quadratic: (2/6)[0 + 4(-1) + 0] = -4/3.

1. 1
   Set up the average

   The interval length is 2.

   $$f_{\mathrm{avg}}=\frac12\int_0^2 3x^2dx$$
2. 2
   Evaluate

   An antiderivative is x cubed.

   $$f_{\mathrm{avg}}=\frac12[ x^3 ]_0^2=4$$

Independent check: One-panel Simpson's rule is exact for 3x²: (2/6)[0 + 4(3) + 12] = 8; dividing by the interval length 2 gives 4.

1. 1
   Identify the ratio

   Each term is one third of the preceding term.

   $$a=1,\quad r=\frac13$$
2. 2
   Check convergence and sum

   The absolute value of r is below one.

   $$S=\frac{1}{1-1/3}=\frac32$$

Independent check: The partial sums 1, 4/3, 13/9, and later values approach 1.5 from below.

1. 1
   Integrate the derivative

   An antiderivative of 3x squared is x cubed.

   $$y=x^3+C$$
2. 2
   Apply the initial condition

   Substitute x = 0 and y = 2.

   $$2=0+C\Rightarrow C=2$$

Independent check: Differentiation gives y′ = 3x², and substitution gives y(0) = 2.