Math 129 Midterm 1 Study Guide
Math 129 Midterm 1 Study Guide Math 129
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This 3 page Study Guide was uploaded by Emma Morrissey on Sunday September 25, 2016. The Study Guide belongs to Math 129 at University of Southern California taught by Takahiro Sakai in Fall 2016. Since its upload, it has received 61 views. For similar materials see Calculus II for Engineers and Scientists in Math at University of Southern California.
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Date Created: 09/25/16
Midterm 1 Study Guide ● Inverse functions (One-to-one) ○ A function has an inverse function if it passes the vertical and horizontal line test ○ Domain and range are switched ○ Reflect across y=x ● Differentiation ○ The derivative is the instantaneous slope/ rate of change/change of rate of a function ○ Differentiation rules ○ Use to find extrema ■ Extrema exist where the first derivative is zero ○ Extreme Value Theorem ■ If the function is continuous and differentiable on a closed interval, it has global extrema ○ Fermat Theorem ■ If local extrema and the first derivative at a point x=C exist, then f’(c) = 0 at the extrema ○ Mean Value Theorem ■ See equation ○ Analyze whether the function is increasing/decreasing and concave up/concave down ■ Increasing when f’(x)>0 ■ Decreasing when f’(x)<0 ■ Concave up when f’’(x)>0 ■ Concave down when f’’(x)<0 ● Inverse Trigonometric Functions ○ Trig functions are only one-to-one on certain intervals π π ■ Sin: − ≤2x ≤ 2 ■ Cos: 0 ≤ x ≤ π π π ■ Tan: − ≤2x ≤ 2 ○ Remember that domain and range are switched for inverses ● Integration ○ Compute the area below a function’s curve ○ Integration Rules ○ Methods ■ Integral Approximation ● Riemann Sums (Right Point, Left Point, Mid Point) ● Error Calculations (See equations) ■ U-Substitution ■ Integration By Parts ■ Partial Fractions ○ Special Integral Scenarios ■ Powers of trig functions ■ Square roots in denominator that can be manipulated into trig functions ○ Evaluation Theorem ○ Fundamental Theorem of Calculus ○ Improper Integrals ■ When the function is not continuous or the limits of integration cannot be evaluated ● Set a dummy variable for the integral and take the limit ● If the limit exists, it is convergent at that point ● If the limit does not exist, it is divergent ■ Power Theorem ■ Comparison Test ○ Calculating area between two curves ■ Find the difference ● Hyperbolic Functions ○ Identities ○ Inverses ● L’Hospital’s Rule ○ When your limit produces an indeterminate quotient, you take the derivative of the numerator and denominator until it can be determined ■ Know what is indeterminate ■ Indeterminate product ● Manipulate into quotient ■ Indeterminate difference ● Manipulate into product ■ Indeterminate powers ● Take the natural log then use e^answer MATH 129 Midterm 1 Equations (f(x) •g(x)) ′= f (x)g(x) +f(x)g (x) cos (x) = (12+cos(2x)) MV T : f (c) = f(b−af(a) sin (x) = (1− cos(2x)) 2 (sin (x)) ′= 1 √1−x 2 ∫tan(x)dx = ln|sec(x)| + C (cos (x)) ′= −1 √1−x 2 ∫sec(x)dx = ln|sec(x)+ tan(x)| + C −1 (tan (x)) ′= 1 2 1+x M =NΔx f(x )[ + 1(x )+ ...2+ f(x N ] b f avg = 1 ∫ f(x)dx) b−a a T N Δ2 [ f(x1)+ 2f(x )2+... + 2f(x N−1 ) +f(x )n ] x −x sinh(x) = e −e Approx Error = |approximation − exact| 2 e +e−x k(b−a)3 cosh(x) = 2 Error Trapezoidal ≤ 12n 2 3 cosh(x) ′ = sinh(x) Error Middle ≤ k(b−2) 24n sinh(x) ′= cosh(x) 2 2 cosh (x)− sinh (x) = 1 ∫udv = uv− vdu∫
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